test_results.json

[
  {
    "test_case": {
      "name": "Arena Bench Hard",
      "system_prompt": "",
      "query": "Write a Python program to build an RL model to recite text from any position that the user provides, using only numpy."
    },
    "results": [
      {
        "approach": "cot_reflection",
        "result": [
          "This Python program defines a reinforcement learning environment for reciting text from a user-defined position using only NumPy. It allows users to navigate through the text, recite characters, and receive rewards based on their recitations.",
          904
        ],
        "time": 10.57605504989624,
        "status": "success"
      },
      {
        "approach": "leap",
        "result": [
          "Building a reinforcement learning (RL) model from scratch using only NumPy can be quite complex, especially if the goal is to recite text from any position. However, I can provide a simplified version that demonstrates the basics of creating a text-reciting scenario with user-input positions using Q-learning in reinforcement learning.\n\nIn this example, the model will learn to recite text one character at a time based on user-provided starting positions. Here\u2019s a basic outline along with the implementation in Python:\n\n### Python Implementation\n\n```python\nimport numpy as np\n\nclass TextReciter:\n    def __init__(self, text):\n        self.text = text\n        self.state_size = len(text)\n        self.action_size = 2  # 0: next character, 1: repeat character\n        self.q_table = np.zeros((self.state_size, self.action_size))\n        self.learning_rate = 0.1\n        self.discount_factor = 0.95\n        self.epsilon = 0.1  # Exploration rate\n\n    def select_action(self, state):\n        if np.random.rand() < self.epsilon:\n            return np.random.choice(self.action_size)  # Explore\n        else:\n            return np.argmax(self.q_table[state])  # Exploit\n\n    def update_q_value(self, state, action, reward, next_state):\n        best_next_action = np.argmax(self.q_table[next_state])\n        td_target = reward + self.discount_factor * self.q_table[next_state][best_next_action]\n        self.q_table[state][action] += self.learning_rate * (td_target - self.q_table[state][action])\n\n    def get_reward(self, action, position):\n        if action == 0:  # Moving to next character\n            if position < self.state_size - 1:\n                return 1  # Positive reward for moving forward\n            else:\n                return -1  # Negative reward for being at the end\n        elif action == 1:  # Repeating current character\n            return 0  # Neutral reward for repeating\n\n    def train(self, episodes):\n        for _ in range(episodes):\n            position = np.random.randint(0, self.state_size)  # Start position\n            done = False\n\n            while not done:\n                state = position\n                action = self.select_action(state)\n\n                if action == 0:  # Move to the next character\n                    next_position = position + 1 if position < self.state_size - 1 else position\n                else:  # Repeat the current character\n                    next_position = position\n\n                reward = self.get_reward(action, position)\n\n                self.update_q_value(state, action, reward, next_position)\n\n                position = next_position\n                done = position >= self.state_size - 1\n\n    def recite(self, start_position):\n        if start_position < 0 or start_position >= self.state_size:\n            print(\"Invalid starting position.\")\n            return\n\n        position = start_position\n        recited_text = \"\"\n\n        while position < self.state_size:\n            recited_text += self.text[position]\n            action = self.select_action(position)\n\n            if action == 0:  # Move to the next character\n                position += 1\n            else:  # Repeat the current character\n                position = position  # No change in position\n\n        return recited_text\n\n\n# Example usage\ntext_to_recite = \"Hello, world!\"\nreciter = TextReciter(text_to_recite)\nreciter.train(1000)  # Train the model on the text\n\n# Recite text starting from a given position\nstarting_position = 7  # Example; user can choose position\nrecited_output = reciter.recite(starting_position)\nprint(\"Recited Text from position {}: {}\".format(starting_position, recited_output))\n```\n\n### Explanation of the Code:\n\n1. **TextReciter Class**: This class encapsulates the behavior of the text-reciting agent. It uses Q-learning to learn how to recite text based on user actions.\n\n2. **Q-table**: This stores the value of state-action pairs, initialized to zeros.\n\n3. **Actions**:\n   - **0**: Move to the next character\n   - **1**: Repeat the current character\n\n4. **Training**: The `train` method allows the agent to explore different states and learn the best actions to take in those states.\n\n5. **Recitation**: The `recite` method takes a starting position, recites the text, and applies the learned Q-values to decide whether to move to the next character or repeat one.\n\nThis is a simplified version of how an RL model might operate for the specific task of reciting text. In practical applications, more sophisticated techniques and libraries (like TensorFlow, PyTorch) would be used for deep reinforcement learning.",
          1004
        ],
        "time": 10.574878931045532,
        "status": "success"
      },
      {
        "approach": "re2",
        "result": [
          "Creating a reinforcement learning (RL) model from scratch using only NumPy to recite text from any starting position as specified by the user is a complex task. However, I can provide a simplified example that illustrates how you can structure such a problem.\n\nIn this context, \"reciting text\" can be interpreted as moving through the text in both directions based on user input. The RL model will be very basic and won't use advanced techniques due to the limitations of the request (i.e., using only NumPy). \n\nHere's a simple implementation:\n\n```python\nimport numpy as np\n\nclass TextReciter:\n    def __init__(self, text):\n        self.text = text\n        self.position = 0  # Starting position in the text\n        self.actions = [\"next\", \"previous\", \"reset\"]  # Possible actions\n\n    def reset(self):\n        self.position = 0  # Reset to the start of the text\n        return self.position\n\n    def move(self, action):\n        if action == \"next\":\n            if self.position < len(self.text) - 1:\n                self.position += 1\n        elif action == \"previous\":\n            if self.position > 0:\n                self.position -= 1\n        elif action == \"reset\":\n            self.reset()\n            \n        return self.position\n\n    def recite(self):\n        return self.text[self.position]\n\n    def available_actions(self):\n        # Returns actions that can be taken based on the current position\n        actions = []\n        if self.position < len(self.text) - 1:\n            actions.append(\"next\")\n        if self.position > 0:\n            actions.append(\"previous\")\n        actions.append(\"reset\")\n        return actions\n\n\ndef main():\n    text = \"This is an example text for recitation using reinforcement learning.\"\n    reciter = TextReciter(text)\n\n    print(\"Initial Text Position: \", reciter.recite())\n\n    while True:\n        print(\"\\nCurrent position: \", reciter.position)\n        print(\"Current character: \", reciter.recite())\n        print(\"Available actions: \", reciter.available_actions())\n        \n        action = input(\"Choose action (next, previous, reset) or 'exit' to quit: \").strip().lower()\n\n        if action == 'exit':\n            break\n        elif action in reciter.available_actions():\n            reciter.move(action)\n        else:\n            print(\"Invalid action! Please choose from available actions.\")\n\n\nif __name__ == \"__main__\":\n    main()\n```\n\n### Explanation:\n- **TextReciter Class**: This class manages the text and allows navigation through its characters.\n- **Reset Method**: Resets the current position to the beginning of the text.\n- **Move Method**: Changes the position based on user actions (`next`, `previous`, `reset`).\n- **Recite Method**: Returns the character at the current position.\n- **Available Actions**: Gives possible actions based on the current position.\n  \n### Running the Program:\n1. Run the program.\n2. You will see the current character and can choose from possible actions (`next`, `previous`, `reset`).\n3. Input your desired action, and the position will be updated accordingly.\n4. Type `exit` to quit the program.\n\n### Note:\nThis example is a very basic demonstration of moving through text based on user input. True reinforcement learning would typically involve a more complex state-action-reward framework, which is not demonstrated here due to simplicity and constraints placing emphasis on using only NumPy.",
          718
        ],
        "time": 12.151690006256104,
        "status": "success"
      },
      {
        "approach": "bon",
        "result": [
          "Creating a reinforcement learning (RL) model to recite text from any position in a given string using only NumPy is an interesting challenge. However, RL typically requires a more comprehensive setup, including an environment, actions, states, rewards, and training loops. \n\nHere's an oversimplified example of how you might implement a simple RL-style model using NumPy, assuming you have a predefined set of possible actions (like moving forward and backward through the text) and you want to train an agent to recite text starting from a user-defined position. \n\nThis example will guide you through a basic structure:\n\n```python\nimport numpy as np\n\nclass ReciterEnv:\n    def __init__(self, text):\n        self.text = text\n        self.position = 0\n        self.done = False\n\n    def reset(self):\n        self.position = 0\n        self.done = False\n        return self.position\n\n    def step(self, action):\n        # action: 0 = forward, 1 = backward, 2 = recite\n        if action == 0:  # Move forward\n            if self.position < len(self.text) - 1:\n                self.position += 1\n        elif action == 1:  # Move backward\n            if self.position > 0:\n                self.position -= 1\n        elif action == 2:  # Recite the text from current position\n            recited_text = self.text[self.position:]\n            self.done = True\n            return recited_text, self.done\n        return self.position, self.done\n\nclass ReciterAgent:\n    def __init__(self, n_actions):\n        self.q_table = np.zeros((len(text), n_actions))  # Q-Table\n        self.epsilon = 0.1  # Exploration probability\n        self.alpha = 0.1   # Learning rate\n        self.gamma = 0.9   # Discount factor\n\n    def choose_action(self, state):\n        if np.random.rand() < self.epsilon:  # Explore\n            return np.random.randint(3)  # Random action\n        else:  # Exploit\n            return np.argmax(self.q_table[state])\n\n    def learn(self, state, action, reward, next_state):\n        best_next_action = np.argmax(self.q_table[next_state])\n        td_target = reward + self.gamma * self.q_table[next_state][best_next_action]\n        self.q_table[state][action] += self.alpha * (td_target - self.q_table[state][action])\n\ndef main():\n    text = \"This is an example text for recital.\"\n    env = ReciterEnv(text)\n    agent = ReciterAgent(n_actions=3)\n\n    episodes = 100  # Number of episodes for training\n\n    for episode in range(episodes):\n        state = env.reset()\n        done = False\n\n        while not done:\n            action = agent.choose_action(state)\n            next_state, done = env.step(action)\n\n            # Define reward: simple reward mechanism, more can be added\n            reward = 0\n            if action == 2 and done:  # If we recited the text\n                reward = 1  # Positive reward for reciting\n\n            agent.learn(state, action, reward, next_state)\n            state = next_state\n\n    # After training, show learned Q-values:\n    print(\"Learned Q-Values:\")\n    print(agent.q_table)\n\n    # Example usage\n    start_position = int(input(\"Start recitation from position (0 to {}): \".format(len(text)-1)))\n    env.reset()\n    env.position = start_position\n\n    while True:\n        action = int(input(\"Choose action (0: forward, 1: backward, 2: recite): \"))\n        if action < 0 or action > 2:\n            print(\"Invalid action.\")\n            continue\n        result, done = env.step(action)\n        if action == 2:\n            print(\"Recited text:\", result)\n            break\n        print(\"Current position:\", env.position)\n\nif __name__ == \"__main__\":\n    main()\n```\n\n### Notes:\n1. **Environment**: The `ReciterEnv` class simulates an environment where an agent can move through the text or recite it.\n2. **Agent**: The `ReciterAgent` manages a Q Table, selects actions based on epsilon-greedy exploration, and updates its Q-values based on rewards.\n3. **Training**: The main loop simulates episodes where the agent interacts with the environment.\n4. **User Interaction**: After training, the user can select the starting position and choose actions to interact with the text.\n\n### Limitations:\n- This is a basic RL setup and might not yield optimal results; real applications often utilize more advanced libraries like TensorFlow or PyTorch for NN-based learning.\n- The reward mechanism is simplistic, and the state-space representation can be made more sophisticated.\n- This code serves an educational purpose and is not fully optimized for performance or functionality.",
          2620
        ],
        "time": 13.767206192016602,
        "status": "success"
      },
      {
        "approach": "z3",
        "result": [
          "Creating a reinforcement learning (RL) model to recite text from any position using only NumPy is a complex task, especially since RL typically involves more sophisticated libraries like TensorFlow or PyTorch for handling neural networks and training processes. However, I can provide a simplified version of a text recitation model using a basic approach with NumPy.\n\nIn this example, we will create a simple environment where the agent learns to recite text from a given position. The agent will receive a reward for reciting the correct character and a penalty for incorrect characters. This is a very basic implementation and does not represent a full RL framework but gives you an idea of how to structure it.\n\nHere's a simple implementation:\n\n```python\nimport numpy as np\n\nclass TextRecitationEnv:\n    def __init__(self, text):\n        self.text = text\n        self.position = 0\n        self.done = False\n\n    def reset(self):\n        self.position = 0\n        self.done = False\n        return self.position\n\n    def step(self, action):\n        if self.done:\n            raise Exception(\"Episode is done. Please reset the environment.\")\n\n        reward = 0\n        if action == self.text[self.position]:\n            reward = 1  # Correct character\n        else:\n            reward = -1  # Incorrect character\n\n        self.position += 1\n        if self.position >= len(self.text):\n            self.done = True\n\n        return self.position, reward, self.done\n\nclass SimpleAgent:\n    def __init__(self, text):\n        self.text = text\n        self.position = 0\n\n    def act(self):\n        if self.position < len(self.text):\n            return self.text[self.position]\n        else:\n            return None  # No more characters to recite\n\ndef train_agent(text, episodes=10):\n    env = TextRecitationEnv(text)\n    agent = SimpleAgent(text)\n\n    for episode in range(episodes):\n        state = env.reset()\n        total_reward = 0\n\n        while True:\n            action = agent.act()\n            if action is None:\n                break\n\n            next_state, reward, done = env.step(action)\n            total_reward += reward\n            agent.position += 1  # Move to the next character\n\n            if done:\n                break\n\n        print(f\"Episode {episode + 1}: Total Reward: {total_reward}\")\n\nif __name__ == \"__main__\":\n    text = \"hello\"\n    train_agent(text, episodes=5)\n```\n\n### Explanation:\n1. **TextRecitationEnv**: This class represents the environment where the agent interacts. It has methods to reset the environment and step through it based on the agent's actions.\n2. **SimpleAgent**: This class represents a very basic agent that recites the text character by character.\n3. **train_agent**: This function runs the training loop for a specified number of episodes. The agent attempts to recite the text, and the environment provides rewards based on the correctness of the actions.\n\n### Limitations:\n- This implementation is very simplistic and does not include any learning mechanism. In a real RL scenario, you would typically use Q-learning or policy gradients to update the agent's policy based on the rewards received.\n- The agent does not learn from its mistakes; it simply recites the text in order.\n- There is no exploration-exploitation strategy implemented.\n\nFor a more sophisticated RL model, consider using libraries like TensorFlow or PyTorch, which provide the necessary tools to build and train neural networks effectively.",
          718
        ],
        "time": 19.07560396194458,
        "status": "success"
      },
      {
        "approach": "moa",
        "result": [
          "Building a reinforcement learning (RL) model from scratch using only NumPy to recite text from a specified position is a challenging yet interesting task. Below is a simplified implementation that incorporates basic RL concepts, such as states, actions, and rewards, while allowing user interaction to recite text.\n\n### Simplified Text Recitation Model\n\nIn this implementation:\n- **State**: The current position in the text.\n- **Actions**: The action is to recite the expected character.\n- **Rewards**: A positive reward for correct recitation and a negative reward for incorrect recitation.\n\nHere's the code:\n\n```python\nimport numpy as np\n\nclass SimpleTextReciter:\n    def __init__(self, text):\n        self.text = text\n        self.position = 0  # Start at the beginning of the text\n        self.q_values = np.zeros(len(text))  # Q-values for each position in the text\n        self.alpha = 0.1  # Learning rate\n        self.gamma = 0.9  # Discount factor\n        self.epsilon = 1.0  # Exploration rate\n        self.epsilon_decay = 0.99  # Decay rate for exploration\n        self.epsilon_min = 0.1  # Minimum exploration rate\n\n    def reset(self):\n        \"\"\"Reset the position to the start of the text.\"\"\"\n        self.position = 0\n\n    def choose_action(self):\n        \"\"\"Choose an action based on epsilon-greedy strategy.\"\"\"\n        if np.random.rand() < self.epsilon:  # Explore\n            return np.random.choice(list(self.text))  # Random character\n        else:  # Exploit\n            return self.text[self.position]  # Expected character\n\n    def update_q_values(self, action, reward):\n        \"\"\"Update Q-values based on the action taken and the received reward.\"\"\"\n        expected_char = self.text[self.position]\n        if action == expected_char:\n            reward = 1  # Positive reward for correct action\n        else:\n            reward = -1  # Negative reward for incorrect action\n\n        # Update Q-value for the current position\n        self.q_values[self.position] += self.alpha * (reward + self.gamma * np.max(self.q_values[self.position + 1:]) - self.q_values[self.position])\n\n    def recite_from_position(self, start_position):\n        \"\"\"Recite text from a given position.\"\"\"\n        self.position = start_position\n        output = ''\n        \n        while self.position < len(self.text):\n            action = self.choose_action()  # Choose an action\n            user_input = input(f\"Current character is '{self.text[self.position]}'. What do you recite? \")\n            \n            self.update_q_values(user_input, action)  # Update Q-values based on user input\n            \n            if user_input == self.text[self.position]:\n                output += user_input  # Append the correct character to the output\n                print(\"Correct! Moving to the next character.\")\n                self.position += 1  # Move to the next character\n            else:\n                print(\"Incorrect. Please try again.\")\n            \n            # Decay epsilon to reduce exploration over time\n            if self.epsilon > self.epsilon_min:\n                self.epsilon *= self.epsilon_decay\n\n        print(\"Recitation complete. You've finished the text!\")\n\ndef main():\n    text = \"Hello, welcome to the text recitation model!\"\n    reciter = SimpleTextReciter(text)\n\n    starting_pos = int(input(\"Enter a starting position (0 to {}): \".format(len(text) - 1)))\n    if 0 <= starting_pos < len(text):\n        reciter.recite_from_position(starting_pos)\n    else:\n        print(\"Invalid starting position.\")\n\nif __name__ == \"__main__\":\n    main()\n```\n\n### Explanation:\n1. **Initialization**: The `SimpleTextReciter` class initializes the text, position, Q-values, learning rate, discount factor, and exploration parameters.\n2. **Choosing Actions**: The `choose_action` method implements an epsilon-greedy strategy to balance exploration and exploitation.\n3. **Updating Q-values**: The `update_q_values` method updates the Q-values based on the action taken and the reward received.\n4. **Recitation**: The `recite_from_position` method allows the user to recite text starting from a specified position, providing feedback on correctness and updating the model accordingly.\n5. **Main Function**: The main function prompts the user for a starting position and initiates the recitation process.\n\n### How to Run:\n1. Copy the code into a Python file (e.g., `text_recitation.py`).\n2. Run the program, provide the starting position, and then try to recite the text by providing the next characters.\n\n### Note:\nThis implementation is a basic demonstration of RL concepts and does not utilize advanced techniques or deep learning. For more complex tasks, consider using libraries like TensorFlow or PyTorch.",
          4215
        ],
        "time": 31.268208742141724,
        "status": "success"
      },
      {
        "approach": "plansearch",
        "result": [
          [
            "```python\nimport numpy as np\n\ndef encode_text(text, position):\n    vocabulary = sorted(list(set(text)))  # Create a vocabulary\n    char_to_index = {char: idx for idx, char in enumerate(vocabulary)}\n    \n    # One-hot encode the text up to the given position\n    one_hot_matrix = np.zeros((len(text), len(vocabulary)))\n    for i, char in enumerate(text):\n        one_hot_matrix[i, char_to_index[char]] = 1\n        \n    # Get the one-hot representation up to the specified position\n    state_representation = one_hot_matrix[:position + 1]  # include position character\n    return state_representation, vocabulary, char_to_index\n\ndef compute_reward(prediction, target):\n    if prediction == target:\n        return 1  # Perfect match\n    elif prediction in target:\n        return 0.5  # Partial match\n    else:\n        return -1  # Incorrect\n\ndef select_action(state, epsilon):\n    if np.random.rand() < epsilon:\n        return np.random.choice(range(state.shape[1]))  # Explore: random choice\n    else:\n        return np.argmax(state[-1])  # Exploit: choose best known action\n\nclass RecitalModel:\n    def __init__(self, vocabulary):\n        self.vocabulary = vocabulary\n        self.char_to_index = {char: idx for idx, char in enumerate(vocabulary)}\n        self.history = []  # Store the history of recited characters\n        \n    def add_to_history(self, character):\n        self.history.append(character)\n        if len(self.history) > 5:  # Keep the last 5 recitations\n            self.history.pop(0)\n\n    def current_state(self, state_representation):\n        # Concatenate history and current state representation\n        return np.concatenate((state_representation.flatten(), self.history_vector()))\n\n    def history_vector(self):\n        history_vector = np.zeros((5, len(self.vocabulary)))\n        for i, char in enumerate(self.history):\n            if i < 5:\n                history_vector[i, self.char_to_index[char]] = 1\n        return history_vector.flatten()  # Flatten for input\n\nclass CharacterModel:\n    def train(self, text_segment):\n        # Placeholder for character model training logic\n        pass\n\nclass PhraseModel:\n    def train(self, text_segment):\n        # Placeholder for phrase model training logic\n        pass\n\nclass HierarchicalModel:\n    def __init__(self):\n        self.character_model = CharacterModel()  # Base model for characters\n        self.phrase_model = PhraseModel()  # Secondary model for phrases\n        \n    def learn(self, text_segment):\n        # Train character level and then phrase level\n        self.character_model.train(text_segment)\n        self.phrase_model.train(text_segment)\n\ndef train_model(text, model, epochs=1000, epsilon=0.1):\n    for epoch in range(epochs):\n        position = np.random.randint(0, len(text))\n        state, vocabulary, char_to_index = encode_text(text, position)\n        char_to_recite = text[position]\n        \n        # Select action and receive prediction\n        action = select_action(state, epsilon)\n        predicted_char = vocabulary[action]  # Convert index back to character\n        reward = compute_reward(predicted_char, char_to_recite)\n        \n        # Update model based on prediction and reward\n        model.learn(state)\n        model.add_to_history(predicted_char)\n\n# Example usage\ntext = \"Hello, Reinforcement Learning!\"\nmodel = HierarchicalModel()\ntrain_model(text, model)\n```"
          ],
          2737
        ],
        "time": 33.406930923461914,
        "status": "success"
      },
      {
        "approach": "rto",
        "result": [
          "```python\nimport numpy as np\n\nclass TextRecitationEnv:\n    def __init__(self, text):\n        self.text = text\n        self.length = len(text)\n        self.state = 0  # Starting position\n\n    def reset(self, start_position):\n        if 0 <= start_position < self.length:\n            self.state = start_position\n        else:\n            raise ValueError(\"Start position out of bounds.\")\n        return self.state\n\n    def step(self, action):\n        # Ensure the action does not exceed the remaining text length\n        if action < 1 or self.state + action > self.length:\n            action = self.length - self.state  # Adjust action to recite remaining text\n\n        recited_text = self.text[self.state:self.state + action]\n        reward = len(recited_text)  # Reward is the length of the recited text\n        self.state += action  # Move the state forward\n\n        done = self.state >= self.length  # Check if the text is fully recited\n        return recited_text, reward, done\n\n\nclass SimpleAgent:\n    def __init__(self, state_size, action_size, epsilon=0.1, alpha=0.1, gamma=0.9):\n        self.q_table = np.zeros((state_size, action_size))  # Initialize Q-table\n        self.epsilon = epsilon  # Exploration rate\n        self.alpha = alpha  # Learning rate\n        self.gamma = gamma  # Discount factor\n\n    def choose_action(self, state):\n        if np.random.rand() < self.epsilon:  # Explore\n            return np.random.randint(1, self.q_table.shape[1])  # Random action\n        else:  # Exploit\n            return np.argmax(self.q_table[state])  # Best action based on Q-table\n\n    def update_q_table(self, state, action, reward, next_state):\n        best_next_action = np.argmax(self.q_table[next_state])\n        td_target = reward + self.gamma * self.q_table[next_state][best_next_action]\n        td_delta = td_target - self.q_table[state][action]\n        self.q_table[state][action] += self.alpha * td_delta  # Update Q-value\n\n\ndef main():\n    text = \"This is a simple text recitation example.\"\n    env = TextRecitationEnv(text)\n    action_size = 10  # Maximum number of characters to recite in one action\n    agent = SimpleAgent(state_size=len(text), action_size=action_size)\n\n    episodes = 1000  # Number of training episodes\n    for episode in range(episodes):\n        start_position = np.random.randint(0, len(text))\n        state = env.reset(start_position)\n\n        done = False\n        while not done:\n            action = agent.choose_action(state)\n            recited_text, reward, done = env.step(action)\n            next_state = env.state\n\n            agent.update_q_table(state, action, reward, next_state)\n            state = next_state\n\n    # Testing the agent\n    print(\"Testing the trained agent:\")\n    start_position = 0  # Start from the beginning for testing\n    state = env.reset(start_position)\n    done = False\n    while not done:\n        action = agent.choose_action(state)\n        recited_text, reward, done = env.step(action)\n        print(f\"Recited: '{recited_text}'\")\n        state = env.state\n\n\nif __name__ == \"__main__\":\n    main()\n```",
          3153
        ],
        "time": 36.79369783401489,
        "status": "success"
      },
      {
        "approach": "mcts",
        "result": [
          "To create a reinforcement learning (RL) model that can recite text from any position provided by the user using only NumPy, we will define a simplified RL framework. This will involve:\n\n1. A text input for our model to recite from.\n2. A state that represents the position in the text.\n3. An action to receive input from the user for a specific position to start reciting from.\n4. A reward system based on the successful recitation of the text from that position.\n\nThis is a minimal implementation and may lack many sophisticated features of RL, but it gives an idea of how you can set up the system. Note that we won't be using any deep learning libraries or advanced RL concepts here, sticking strictly to NumPy.\n\nHere\u2019s how we can implement this:\n\n```python\nimport numpy as np\n\nclass TextReciterRL:\n    def __init__(self, text):\n        self.text = text\n        self.state = 0  # Current position in the text\n\n    def reset(self):\n        self.state = 0  # Reset position to start\n\n    def get_state(self):\n        return self.state\n\n    def recite(self, position):\n        # Check if the position is valid\n        if position < 0 or position >= len(self.text):\n            print(\"Invalid position. Please enter a position between 0 and\", len(self.text)-1)\n            return None\n        \n        # Move the state to the given position and recite\n        self.state = position\n        recited_text = self.text[self.state:]\n        \n        print(\"Reciting from position\", self.state, \":\", recited_text)\n        return recited_text\n\n    def step(self, position):\n        \"\"\"Take a step in the environment by trying to recite from a given position.\"\"\"\n        reward = 0\n        recited_text = self.recite(position)\n        \n        if recited_text is not None:\n            # Simple reward system: reward is 1 if we recite successfully\n            reward = 1\n        return reward\n\ndef main():\n    # Example text\n    text = \"Reinforcement learning is a type of machine learning that focuses on training agents to make a sequence of decisions.\"\n\n    # Create an instance of the TextReciterRL\n    reciter = TextReciterRL(text)\n\n    # Example user input: position to start reciting from\n    while True:\n        user_input = input(\"Enter a position to recite from (or 'exit' to stop): \")\n        \n        if user_input.lower() == 'exit':\n            break\n        \n        try:\n            position = int(user_input)\n            reward = reciter.step(position)\n            print(\"Received reward:\", reward)\n        except ValueError:\n            print(\"Please enter a valid integer.\")\n\nif __name__ == \"__main__\":\n    main()\n```\n\n### Explanation:\n\n- **Initialization**: The `TextReciterRL` class initializes with the text to recite from.\n- **State Management**: The current position in the text (state) is stored and can be reset.\n- **Recitation**: The `recite` method checks if the provided position is valid and returns the text from that position; if valid, it recites the text from `self.state` onward.\n- **Reward System**: In the `step` method, we compute a reward based on whether the recitation occurred successfully. A simplistic reward of `1` is given whenever a successful recitation occurs.\n- **User Interaction**: In the `main` function, we allow the user to input positions to recite from, with the option to exit the program.\n\nThis simple program illustrates concepts of state representation and rewards but does not include training or learning mechanisms, which are core to reinforcement learning. A complete RL implementation would involve using states, actions, rewards, and Q-values or neural networks for training, which is more complex and typically requires additional libraries like TensorFlow or PyTorch for effectiveness in real-world applications. However, this example follows the constraints provided about using only NumPy.",
          10854
        ],
        "time": 54.5164008140564,
        "status": "success"
      },
      {
        "approach": "self_consistency",
        "result": [
          "Building a Reinforcement Learning (RL) model that can recite text based on a user-provided position is quite complex and typically requires advanced frameworks such as TensorFlow or PyTorch. However, I can provide a simplified version using just `numpy` which mimics an RL-style approach but with far less complexity. \n\nThis example won't strictly adhere to RL principles, but will illustrate how you could create an environment where you can recite text from a given position using basic numpy functionalities. Here, I\u2019m focusing on a simple approach to handle text recitation.\n\nPlease note that this code is not an RL model in the true sense (no agents, rewards, etc.), but it demonstrates basic functionalities without using advanced libraries.\n\nHere's a simple textual recitation program:\n\n```python\nimport numpy as np\n\nclass TextReciter:\n    def __init__(self, text):\n        # Store the text\n        self.text = text\n        self.position = 0  # starting position\n\n    def get_text(self):\n        # Return text from the current position to the end\n        return self.text[self.position:]\n\n    def move_to_position(self, pos):\n        # Move to a new position if it's within range\n        if 0 <= pos < len(self.text):\n            self.position = pos\n        else:\n            print(f\"Position {pos} out of range. Please select a position between 0 and {len(self.text) - 1}.\")\n\n    def recite(self):\n        # Recite the text from the current position\n        output = self.get_text()\n        print(output)\n\n\ndef main():\n    # Sample text to recite\n    sample_text = \"In the beginning, there was light. The light illuminated the darkness.\"\n    \n    # Create a TextReciter instance\n    reciter = TextReciter(sample_text)\n    \n    while True:\n        print(\"\\nCurrent text position:\", reciter.position)\n        print(\"Current text:\", reciter.get_text())\n        user_input = input(\"Enter a position to recite from (or 'exit' to quit): \")\n        \n        if user_input.lower() == 'exit':\n            break\n        \n        try:\n            position = int(user_input)\n            reciter.move_to_position(position)\n            reciter.recite()\n        except ValueError:\n            print(\"Please enter a valid integer or 'exit' to quit.\")\n\nif __name__ == \"__main__\":\n    main()\n```\n\n### Explanation:\n\n1. **Class Definition:** The `TextReciter` class holds the text and the current position. It has methods to get text from the current position, move the position, and recite the text.\n\n2. **Methods:**\n   - `get_text`: Retrieves the text starting from the current position.\n   - `move_to_position`: Validates and sets the current position.\n   - `recite`: Outputs the text from the current position.\n\n3. **Main Loop:** The `main` function initializes the text reciter and enters a loop where the user can enter a position to start reciting from or exit the program.\n\n### Execution:\nTo run this, simply paste the code into a Python environment and execute it. You can enter different text positions to recite the text accordingly.\n\n### Limitations:\nThis approach doesn\u2019t implement any learning aspect nor does it use any RL techniques, which would require reward systems and possibly neural networks. If you are interested in building a more sophisticated RL agent, consider using libraries like TensorFlow or PyTorch, which are designed for those purposes.",
          4348
        ],
        "time": 61.16654586791992,
        "status": "success"
      },
      {
        "approach": "rstar",
        "result": [
          "0",
          12731
        ],
        "time": 72.11845803260803,
        "status": "success"
      },
      {
        "approach": "pvg",
        "result": [
          "To create a basic reinforcement learning (RL) model that can recite text from any position provided by the user using only NumPy, we can design a simple environment and an agent. The agent will learn to output text from a given starting position through reinforcement learning. \n\nFor this example, we will simplify the problem. The agent will learn to recite text based on a reward structure that encourages it to get closer to the desired output. The text will be represented by a sequence of characters, and the agent will take actions based on the current position in the text.\n\nBelow is a step-by-step implementation in Python using NumPy:\n\n### Step 1: Set Up the Environment\n\nWe will create a simple environment that represents the text and allows the agent to take actions based on its current position. The text will be a string, and the agent will be able to move left or right.\n\n### Step 2: Define the Agent\n\nThe agent will be defined to take actions based on its current state (position in the text) and learn from rewards received after each action.\n\n### Step 3: Implement the Learning Process\n\nWe'll use a simple Q-learning algorithm to allow the agent to learn from its experiences.\n\n### Step 4: Combine Everything\n\nHere's how to implement the above steps in Python:\n\n```python\nimport numpy as np\n\nclass TextEnvironment:\n    def __init__(self, text):\n        self.text = text\n        self.length = len(text)\n        self.position = 0  # Start at the beginning of the text\n\n    def reset(self):\n        self.position = 0  # Reset to the start\n        return self.position\n\n    def step(self, action):\n        if action == 0:  # Move left\n            self.position = max(0, self.position - 1)\n        elif action == 1:  # Move right\n            self.position = min(self.length - 1, self.position + 1)\n        \n        # Reward is 1 if the agent is at the last position, else 0\n        reward = 1 if self.position == self.length - 1 else 0\n        return self.position, reward\n\nclass QLearningAgent:\n    def __init__(self, state_size, action_size, learning_rate=0.1, discount_factor=0.9):\n        self.state_size = state_size\n        self.action_size = action_size\n        self.learning_rate = learning_rate\n        self.discount_factor = discount_factor\n        self.q_table = np.zeros((state_size, action_size))  # Initialize Q-table\n\n    def choose_action(self, state):\n        return np.argmax(self.q_table[state])  # Choose the action with the highest Q-value\n\n    def learn(self, state, action, reward, next_state):\n        best_next_action = np.argmax(self.q_table[next_state])\n        td_target = reward + self.discount_factor * self.q_table[next_state][best_next_action]\n        td_delta = td_target - self.q_table[state][action]\n        self.q_table[state][action] += self.learning_rate * td_delta  # Update Q-value\n\n# Initialize environment and agent\ntext = \"Hello, this is a simple text.\"\nenv = TextEnvironment(text)\nagent = QLearningAgent(state_size=len(text), action_size=2)\n\n# Training the agent\nnum_episodes = 1000\nfor episode in range(num_episodes):\n    state = env.reset()\n    done = False\n    while not done:\n        action = agent.choose_action(state)\n        next_state, reward = env.step(action)\n        agent.learn(state, action, reward, next_state)\n        state = next_state\n        if reward == 1:  # Reached end of text\n            done = True\n\n# Recite text from a given position\ndef recite_text_from_position(start_position):\n    if start_position < 0 or start_position >= len(text):\n        return \"Invalid start position.\"\n    \n    current_position = start_position\n    recited_text = \"\"\n    \n    while current_position < len(text):\n        recited_text += text[current_position]\n        current_position += 1  # Move to the next character\n    \n    return recited_text\n\n# Example usage\nstart_position = 7\nprint(f\"Starting from position {start_position}: '{recite_text_from_position(start_position)}'\")\n```\n\n### Explanation of the Code:\n\n1. **TextEnvironment Class**: This class manages the state of the text and allows the agent to take actions (move left or right).\n2. **QLearningAgent Class**: This class implements the Q-learning algorithm. It has methods to choose actions based on the Q-table and to learn from experiences.\n3. **Training Loop**: The agent is trained over multiple episodes to learn the best actions to take in order to reach the end of the text.\n4. **Recite Function**: After training, you can provide a starting position to recite the text from that position.\n\nThis implementation is a simplified example of how to create a text-receiving agent using reinforcement learning concepts with NumPy.",
          16630
        ],
        "time": 114.87957096099854,
        "status": "success"
      }
    ]
  },
  {
    "test_case": {
      "name": "Big Code Bench",
      "system_prompt": "",
      "query": "Calculates the average of the sums of absolute differences between each pair of consecutive numbers for all permutations of a given list. Each permutation is shuffled before calculating the differences. Args: - numbers (list): A list of numbers. Default is numbers from 1 to 10. The function should output with: float: The average of the sums of absolute differences for each shuffled permutation of the list. You should write self-contained code starting with: ``` import itertools from random import shuffle def task_func(numbers=list(range(1, 3))): ```"
    },
    "results": [
      {
        "approach": "leap",
        "result": [
          "Sure! Below is a self-contained Python function that calculates the average of the sums of absolute differences between each pair of consecutive numbers for all permutations of a given list. Each permutation is shuffled before calculating the differences.\n\n```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 3))):\n    total_sum_of_differences = 0\n    total_permutations = 0\n    \n    # Generate all permutations of the input numbers\n    permutations = list(itertools.permutations(numbers))\n    \n    for perm in permutations:\n        # Shuffle the permutation\n        shuffled_perm = list(perm)\n        shuffle(shuffled_perm)\n        \n        # Calculate the sum of absolute differences for the shuffled permutation\n        sum_of_differences = sum(abs(shuffled_perm[i] - shuffled_perm[i + 1]) for i in range(len(shuffled_perm) - 1))\n        \n        total_sum_of_differences += sum_of_differences\n        total_permutations += 1\n    \n    # Calculate the average\n    average = total_sum_of_differences / total_permutations if total_permutations > 0 else 0\n    return average\n\n# Example usage:\nprint(task_func(list(range(1, 11))))  # Calculate the average for numbers from 1 to 10\n```\n\n### Explanation:\n\n1. **Import Statements**: The code imports `itertools` for generating permutations and `shuffle` for shuffling the lists.\n\n2. **Function Definition**: The function `task_func` is defined to take a list of numbers, with the default being numbers from 1 to 2.\n\n3. **Permutation Generation**: The code generates all permutations of the input list using `itertools.permutations`.\n\n4. **Shuffling and Diff Calculation**:\n   - Each permutation is converted to a list and shuffled.\n   - The sum of absolute differences between consecutive elements in the shuffled permutation is calculated.\n\n5. **Average Calculation**: Finally, the average of these sums is computed and returned, handling the case where there are no permutations (though unlikely with the provided input).\n\nYou can test this function with different lists, and it will calculate the required average based on shuffled versions of each permutation.",
          467
        ],
        "time": 5.271653890609741,
        "status": "success"
      },
      {
        "approach": "re2",
        "result": [
          "Here is a self-contained Python function that meets the requirements you've specified. The function calculates the average of the sums of absolute differences between each pair of consecutive numbers for all permutations of a given list. Each permutation is shuffled before calculating the differences.\n\n```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    total_sum_of_differences = 0\n    count = 0\n\n    # Generate all permutations of the list\n    permutations = list(itertools.permutations(numbers))\n    \n    for perm in permutations:\n        # Shuffle the permutation\n        perm_list = list(perm)\n        shuffle(perm_list)\n        \n        # Calculate the sum of absolute differences\n        sum_of_differences = sum(abs(perm_list[i] - perm_list[i + 1]) for i in range(len(perm_list) - 1))\n        \n        total_sum_of_differences += sum_of_differences\n        count += 1\n\n    # Calculate and return the average\n    average_differences = total_sum_of_differences / count\n    return average_differences\n\n# Example usage:\naverage_result = task_func()\nprint(f\"The average of the sums of absolute differences is: {average_result}\")\n```\n\n### Explanation:\n1. **Imports**: The function uses `itertools` to generate permutations and `random.shuffle` to shuffle them.\n2. **Function Definition**: `task_func` takes `numbers`, defaulting to the list from 1 to 10.\n3. **Generate Permutations**: It creates all permutations of the input list.\n4. **Shuffling**: Each permutation is shuffled before calculating the absolute differences.\n5. **Calculate Absolute Differences**: For each shuffled list, it computes the sum of the absolute differences between consecutive elements.\n6. **Average Calculation**: Finally, it calculates and returns the average of these sums.\n\nThis implementation adheres to the specified requirements and will give you the average of the sums of absolute differences for the shuffled permutations.",
          418
        ],
        "time": 5.527898073196411,
        "status": "success"
      },
      {
        "approach": "cot_reflection",
        "result": [
          "```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    total_sum = 0\n    count = 0\n    \n    # Generate all permutations of the original list\n    all_permutations = list(itertools.permutations(numbers))\n    \n    for perm in all_permutations:\n        # Convert the tuple to a list and shuffle it\n        shuffled_perm = list(perm)\n        shuffle(shuffled_perm)\n        \n        # Calculate the sum of absolute differences\n        abs_diff_sum = sum(abs(shuffled_perm[i] - shuffled_perm[i + 1]) for i in range(len(shuffled_perm) - 1))\n        \n        total_sum += abs_diff_sum\n        count += 1\n    \n    # Compute the average\n    average = total_sum / count if count > 0 else 0\n    return average\n```",
          543
        ],
        "time": 7.487663745880127,
        "status": "success"
      },
      {
        "approach": "z3",
        "result": [
          "Certainly! Below is a self-contained Python function that calculates the average of the sums of absolute differences between each pair of consecutive numbers for all permutations of a given list. Each permutation is shuffled before calculating the differences.\n\n```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    total_sum_of_differences = 0\n    count = 0\n    \n    # Generate all permutations of the numbers\n    permutations = list(itertools.permutations(numbers))\n    \n    for perm in permutations:\n        # Shuffle the permutation\n        perm_list = list(perm)\n        shuffle(perm_list)\n        \n        # Calculate the sum of absolute differences for the shuffled permutation\n        sum_of_differences = sum(abs(perm_list[i] - perm_list[i + 1]) for i in range(len(perm_list) - 1))\n        \n        total_sum_of_differences += sum_of_differences\n        count += 1\n    \n    # Calculate the average\n    average = total_sum_of_differences / count if count > 0 else 0\n    return average\n\n# Example usage\nresult = task_func()\nprint(result)\n```\n\n### Explanation:\n1. **Imports**: The code imports `itertools` for generating permutations and `shuffle` from `random` for shuffling.\n2. **Function Definition**: The function `task_func` takes a list of numbers, defaulting to numbers from 1 to 10.\n3. **Permutations**: It generates all permutations of the input list.\n4. **Shuffling and Calculating Differences**: For each permutation, it shuffles the order and calculates the sum of absolute differences between consecutive elements.\n5. **Average Calculation**: It keeps track of the total sum of differences and the count of permutations to compute the average.\n6. **Return Value**: Finally, it returns the average of the sums of absolute differences.\n\nYou can call `task_func()` to see the result for the default list of numbers from 1 to 10.",
          422
        ],
        "time": 8.969255924224854,
        "status": "success"
      },
      {
        "approach": "bon",
        "result": [
          "Below is the self-contained code that calculates the average of the sums of absolute differences between each pair of consecutive numbers for all permutations of a given list. Each permutation is shuffled before calculating the differences.\n\n```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    # Generate all permutations of the numbers\n    permutations = list(itertools.permutations(numbers))\n    \n    total_sum_abs_diff = 0\n    total_permutations = 0\n    \n    # Loop through each permutation\n    for perm in permutations:\n        # Convert permutation to a list and shuffle it\n        shuffled_perm = list(perm)\n        shuffle(shuffled_perm)\n\n        # Compute the sum of absolute differences in the shuffled permutation\n        sum_abs_diff = sum(abs(shuffled_perm[i] - shuffled_perm[i + 1]) for i in range(len(shuffled_perm) - 1))\n\n        total_sum_abs_diff += sum_abs_diff\n        total_permutations += 1\n        \n    # Calculate the average\n    if total_permutations > 0:\n        average_abs_diff = total_sum_abs_diff / total_permutations\n    else:\n        return 0.0  # Avoid division by zero\n\n    return average_abs_diff\n\n# Example usage:\naverage_result = task_func()\nprint(f\"The average of the sums of absolute differences is: {average_result:.2f}\")\n```\n\n### Explanation:\n1. **Imports**: We import `itertools` for generating permutations and `shuffle` from the `random` module to randomize each permutation.\n2. **Function Definition**: `task_func` accepts a list of numbers with a default from 1 to 10.\n3. **Generate Permutations**: We generate all permutations of the input list using `itertools.permutations`.\n4. **Initialize Sums**: We initialize variables to keep track of the total sum of absolute differences and the count of permutations.\n5. **Loop Through Permutations**: For each permutation:\n   - Convert it to a list and shuffle it.\n   - Calculate the sum of absolute differences between consecutive elements.\n6. **Calculate Average**: Finally, we compute the average of these sums and return it.\n7. **Example Usage**: The function can be called to compute and print the average for the default list.\n\nYou can test the function with a different list of numbers by passing a different argument to `task_func()`.",
          1349
        ],
        "time": 11.455209970474243,
        "status": "success"
      },
      {
        "approach": "moa",
        "result": [
          "Here\u2019s an optimized and self-contained Python function that calculates the average of the sums of absolute differences between each pair of consecutive numbers for all permutations of a given list. Each permutation is shuffled before calculating the differences:\n\n```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    total_difference_sum = 0\n    total_permutations = 0\n    \n    # Generate all permutations of the numbers\n    permutations = list(itertools.permutations(numbers))\n    \n    for permutation in permutations:\n        # Shuffle the current permutation\n        shuffled_permutation = list(permutation)\n        shuffle(shuffled_permutation)\n        \n        # Calculate the sum of absolute differences\n        difference_sum = sum(abs(shuffled_permutation[i] - shuffled_permutation[i + 1]) for i in range(len(shuffled_permutation) - 1))\n        \n        total_difference_sum += difference_sum\n        total_permutations += 1\n    \n    # Calculate and return the average\n    average_difference_sum = total_difference_sum / total_permutations if total_permutations > 0 else 0.0\n    return average_difference_sum\n\n# Example usage:\nif __name__ == \"__main__\":\n    average_diff = task_func()\n    print(f\"Average of the sums of absolute differences: {average_diff}\")\n```\n\n### Explanation:\n1. **Imports**: The code imports `itertools` for generating permutations and `shuffle` from `random` for shuffling.\n2. **Function Definition**: The `task_func` function takes a list of numbers, defaulting to numbers from 1 to 10.\n3. **Generating Permutations**: It generates all possible permutations of the input list using `itertools.permutations`.\n4. **Shuffling and Calculating Differences**: For each permutation:\n   - It converts the permutation tuple to a list and shuffles it.\n   - It calculates the sum of absolute differences between each pair of consecutive numbers in the shuffled list.\n5. **Average Calculation**: After iterating through all permutations, it computes the average of the sums of absolute differences.\n6. **Return Value**: The function returns the average, which is printed when the function is called.\n\n### Note:\nWhile the shuffling step is included as per the original request, it does not affect the final result since the sum of absolute differences for a given permutation remains the same regardless of the order. Thus, the shuffling step can be considered unnecessary for the final calculation of the average.",
          2286
        ],
        "time": 17.519530057907104,
        "status": "success"
      },
      {
        "approach": "plansearch",
        "result": [
          [
            "```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    n = len(numbers)\n    if n <= 1:\n        return 0.0\n    \n    total_difference_sum = 0\n    \n    # Iterate through all permutations of the numbers\n    for _ in range(10000):  # Shuffle a limited number of times to approximate\n        perm = numbers[:]\n        shuffle(perm)\n        \n        # Calculate the sum of absolute differences for this permutation\n        current_sum = sum(abs(perm[i] - perm[i + 1]) for i in range(n - 1))\n        total_difference_sum += current_sum\n    \n    # Calculate the average\n    average_sum_of_abs_diffs = total_difference_sum / 10000\n    \n    return average_sum_of_abs_diffs\n\n# Testing the function\nprint(task_func())  # Average of sums of absolute differences for the default list from 1 to 10\n```"
          ],
          1713
        ],
        "time": 19.68016219139099,
        "status": "success"
      },
      {
        "approach": "rto",
        "result": [
          "```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    total_sum_of_differences = 0\n    count = 0\n    \n    # Generate all permutations of the numbers\n    for perm in itertools.permutations(numbers):\n        # Convert the permutation to a list and shuffle it\n        perm_list = list(perm)\n        shuffle(perm_list)\n        \n        # Calculate the sum of absolute differences for the shuffled permutation\n        sum_of_differences = sum(abs(perm_list[i] - perm_list[i + 1]) for i in range(len(perm_list) - 1))\n        \n        total_sum_of_differences += sum_of_differences\n        count += 1\n    \n    # Calculate the average\n    average = total_sum_of_differences / count if count > 0 else 0\n    return average\n\n# Example usage\nresult = task_func()\nprint(\"Average of sums of absolute differences:\", result)\n```",
          1513
        ],
        "time": 24.09026026725769,
        "status": "success"
      },
      {
        "approach": "mcts",
        "result": [
          "Below is the self-contained code that calculates the average of the sums of absolute differences between each pair of consecutive numbers for all permutations of a given list. The list is shuffled before calculating the differences.\n\n```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    total_sum_diff = 0\n    total_permutations = 0\n    \n    # Generate all permutations of the input numbers\n    for perm in itertools.permutations(numbers):\n        # Shuffle the permutation to simulate the shuffling requirement\n        perm_list = list(perm)\n        shuffle(perm_list)\n        \n        # Calculate the sum of absolute differences for the shuffled permutation\n        sum_diff = sum(abs(perm_list[i] - perm_list[i + 1]) for i in range(len(perm_list) - 1))\n        \n        total_sum_diff += sum_diff\n        total_permutations += 1\n        \n    # Calculate the average of the sums of absolute differences\n    average_sum_diff = total_sum_diff / total_permutations if total_permutations > 0 else 0\n    return average_sum_diff\n\n# Example usage\nresult = task_func()\nprint(result)\n```\n\n### Explanation:\n\n1. **Import Statements**: The code imports `itertools` for generating permutations and `shuffle` from the `random` module for shuffling.\n\n2. **Function Definition**: `task_func` is defined with a default argument as numbers ranging from 1 to 10.\n\n3. **Initialization**: It initializes `total_sum_diff` to accumulate the sums of absolute differences and `total_permutations` to count the number of permutations processed.\n\n4. **Permutation Generation**: Using `itertools.permutations`, the function iterates over each permutation of the provided numbers.\n\n5. **Shuffling**: Each permutation is converted to a list and shuffled.\n\n6. **Sum of Absolute Differences**: The sum of absolute differences between each consecutive pair in the shuffled list is calculated and added to `total_sum_diff`.\n\n7. **Calculating Average**: After processing all permutations, it calculates the average by dividing the total sum of differences by the number of permutations.\n\n8. **Return Value**: The function returns the average of the sums of absolute differences. \n\n### Note:\nDue to the nature of permutations and the shuffling, the output might vary with each run, and it's designed to illustrate how to perform the task requested.",
          7158
        ],
        "time": 35.02100205421448,
        "status": "success"
      },
      {
        "approach": "self_consistency",
        "result": [
          "Certainly! Below is a self-contained Python function that computes the average of the sums of absolute differences between each pair of consecutive numbers for all shuffled permutations of a given list:\n\n```python\nimport itertools\nfrom random import shuffle\n\ndef task_func(numbers=list(range(1, 11))):\n    total_sum_of_abs_diff = 0\n    count_of_permutations = 0\n\n    # Generate all unique permutations of the list\n    for perm in set(itertools.permutations(numbers)):\n        # Calculate the sum of absolute differences for the current permutation\n        sum_of_abs_diff = 0\n        for i in range(len(perm) - 1):\n            sum_of_abs_diff += abs(perm[i] - perm[i + 1])\n        \n        total_sum_of_abs_diff += sum_of_abs_diff\n        count_of_permutations += 1\n\n    # Calculate the average\n    if count_of_permutations > 0:\n        average_abs_diff = total_sum_of_abs_diff / count_of_permutations\n    else:\n        average_abs_diff = 0.0\n\n    return average_abs_diff\n\n# Example usage\nresult = task_func()\nprint(\"Average of sums of absolute differences:\", result)\n```\n\n### Explanation of the code:\n1. We import the necessary libraries: `itertools` for permutations and `shuffle` for potential future use (though the current implementation does not use shuffling).\n2. The function `task_func` takes a list of numbers (defaults to `[1, 2, ..., 10]`).\n3. We initialize `total_sum_of_abs_diff` for accumulating the total sums of absolute differences and `count_of_permutations` for counting the number of permutations processed.\n4. We generate all unique permutations of the provided list and calculate the sum of absolute differences for each permutation.\n5. We sum up these differences for all permutations and count their total.\n6. Finally, we calculate and return the average of these sums. If no permutations exist (which won't happen here), we return `0.0`. \n\nFeel free to modify the input list in the function call to test it with different sets of numbers.",
          2183
        ],
        "time": 36.24512314796448,
        "status": "success"
      },
      {
        "approach": "rstar",
        "result": [
          "1",
          13201
        ],
        "time": 68.87629103660583,
        "status": "success"
      },
      {
        "approach": "pvg",
        "result": [
          "To create a function that calculates the average of the sums of absolute differences between each pair of consecutive numbers for a specified number of randomly shuffled permutations of a given list, we will follow these steps:\n\n1. **Import Necessary Libraries**: We will use `random.shuffle()` to shuffle the list and `itertools` if needed for permutations, though it may not be necessary in this context.\n\n2. **Define the Function**: The function will accept a list of numbers and a number indicating how many times to shuffle the list.\n\n3. **Handle Edge Cases**: We will check if the length of the list is less than 2 and return 0.0 since we cannot calculate differences in such cases.\n\n4. **Calculate Absolute Differences**: For each shuffled version of the list, we will compute the sum of absolute differences between consecutive elements.\n\n5. **Compute the Average**: Finally, we will compute the average of these sums and return it as a float.\n\nHere is the implementation of the solution:\n\n```python\nimport random\n\ndef task_func(numbers=list(range(1, 11)), num_shuffles=1000):\n    # Handle edge cases\n    if len(numbers) < 2:\n        return 0.0  # No pairs to compare\n    \n    total_sum_of_differences = 0\n\n    for _ in range(num_shuffles):\n        # Shuffle the list\n        shuffled_numbers = numbers[:]\n        random.shuffle(shuffled_numbers)\n        \n        # Calculate the sum of absolute differences\n        sum_of_differences = sum(abs(shuffled_numbers[i] - shuffled_numbers[i + 1]) for i in range(len(shuffled_numbers) - 1))\n        total_sum_of_differences += sum_of_differences\n    \n    # Calculate the average\n    average_difference = total_sum_of_differences / num_shuffles\n    return float(average_difference)\n\n# Example usage:\nresult = task_func()\nprint(result)\n```\n\n### Explanation of the Code:\n\n1. **Function Definition**: The `task_func` function is defined with two parameters: `numbers`, which defaults to a list from 1 to 10, and `num_shuffles`, which defaults to 1000.\n\n2. **Edge Case Handling**: If the length of `numbers` is less than 2, the function immediately returns `0.0`.\n\n3. **Shuffling and Calculation**:\n   - The function enters a loop that runs `num_shuffles` times.\n   - Each time, it makes a copy of the original list, shuffles it, and calculates the sum of absolute differences between consecutive elements using a generator expression.\n   - This sum is accumulated in `total_sum_of_differences`.\n\n4. **Average Calculation**: After all shuffles, the average is computed by dividing `total_sum_of_differences` by `num_shuffles`, and the result is returned as a float.\n\nThis function effectively meets the requirements laid out in the refined query, handling edge cases and ensuring a robust calculation of the average of the sums of absolute differences.",
          27116
        ],
        "time": 102.12563920021057,
        "status": "success"
      }
    ]
  },
  {
    "test_case": {
      "name": "Maths Problem",
      "system_prompt": "You are a helpful AI assistant with expertise in mathematical and logical problem-solving.",
      "query": "Maximize x + y subject to: x + 2y <= 10, x >= 0, y >= 0"
    },
    "results": [
      {
        "approach": "leap",
        "result": [
          "To maximize the objective function \\( z = x + y \\) subject to the constraints \\( x + 2y \\leq 10 \\), \\( x \\geq 0 \\), and \\( y \\geq 0 \\), we can follow these steps:\n\n1. **Identify the constraints**: \n   - The constraint \\( x + 2y \\leq 10 \\) can be rewritten as \\( y \\leq 5 - \\frac{x}{2} \\).\n   - The constraints \\( x \\geq 0 \\) and \\( y \\geq 0 \\) indicate that we are working in the first quadrant of the Cartesian plane.\n\n2. **Graph the constraints**: \n   - First, graph the line \\( x + 2y = 10 \\). When \\( x = 0 \\), \\( y = 5 \\) (point (0, 5)). When \\( y = 0 \\), \\( x = 10 \\) (point (10, 0)).\n   - Plot these points on a graph and draw a line through them. The area below this line (including the line) corresponds to the feasible region defined by the constraint \\( x + 2y \\leq 10 \\).\n\n3. **Find the vertices of the feasible region**:\n   - The feasible region is bounded by the constraints and will typically be a polygon. The vertices of this region can be determined by the intersections of the lines (constraints) presented and the axes.\n   - The vertices of interest are:\n     1. (0, 0) \u2013 Intersection of the axes\n     2. (0, 5) \u2013 Where \\( x = 0 \\)\n     3. (10, 0) \u2013 Where \\( y = 0 \\)\n     4. Intersection of \\( x + 2y = 10 \\) with the axes, which we already found.\n\n4. **Evaluate the objective function at the vertices**:\n   - At (0, 0): \\( z = 0 + 0 = 0 \\)\n   - At (0, 5): \\( z = 0 + 5 = 5 \\)\n   - At (10, 0): \\( z = 10 + 0 = 10 \\)\n   - We can also find the point of intersection with the vertical axis which is already addressed.\n\n5. **Select the maximum value**:\n   - From the evaluated values, the maximum value of \\( z \\) occurs at the points (10, 0) and (0, 5), but here we can consider how both constraints matter as a balanced approach.\n   - Hence we will consider optimal boundaries that would strike a maximum, which generally falls ideally maximized along axes not out of bounds.\n\nIn conclusion, the maximum value of \\( x + y \\) occurs at the vertex (10, 0) with a maximum of \\( z = 10 \\) or at (0, 5): along constraint cuts of y, while feasible values suggest leading maximum outputs might lie along maximizing across both axes if given additional setup. Therefore, \\( \\boxed{10} \\) is the maximum value.",
          665
        ],
        "time": 5.978002309799194,
        "status": "success"
      },
      {
        "approach": "z3",
        "result": [
          "The optimal solution to the problem of maximizing \\( x + y \\) subject to the constraints \\( x + 2y \\leq 10 \\), \\( x \\geq 0 \\), and \\( y \\geq 0 \\) is:\n\n- \\( x = 10 \\)\n- \\( y = 0 \\)\n- The maximum value of \\( x + y = 10 + 0 = 10 \\).\n\nThus, the maximum value of the objective function \\( x + y \\) is **10**.",
          108
        ],
        "time": 6.33604097366333,
        "status": "success"
      },
      {
        "approach": "cot_reflection",
        "result": [
          "The maximum value of \\( x + y \\) is 10, occurring at the point (10, 0).",
          673
        ],
        "time": 6.999939203262329,
        "status": "success"
      },
      {
        "approach": "re2",
        "result": [
          "To maximize the expression \\( x + y \\) subject to the constraints \\( x + 2y \\leq 10 \\), \\( x \\geq 0 \\), and \\( y \\geq 0 \\), we can use the method of linear programming. \n\n**Step 1: Identify the feasible region.**\n\n1. The line \\( x + 2y = 10 \\) can be rearranged to find the intercepts:\n   - When \\( x = 0 \\): \n     \\[\n     0 + 2y = 10 \\implies y = 5 \n     \\]\n   - When \\( y = 0 \\): \n     \\[\n     x + 0 = 10 \\implies x = 10 \n     \\]\n   - The line intersects the x-axis at \\( (10, 0) \\) and the y-axis at \\( (0, 5) \\).\n\n2. The feasible region is bounded by the constraints:\n   - The line \\( x + 2y = 10 \\)\n   - The x-axis (\\( y = 0 \\))\n   - The y-axis (\\( x = 0 \\))\n\n**Step 2: Determine the corner points of the feasible region.**\n\nThe feasible region is a triangle with the vertices:\n- \\( (0, 0) \\)\n- \\( (10, 0) \\)\n- \\( (0, 5) \\)\n\n**Step 3: Evaluate the objective function \\( x + y \\) at each corner point.**\n\n1. At \\( (0, 0) \\):\n   \\[\n   x + y = 0 + 0 = 0\n   \\]\n\n2. At \\( (10, 0) \\):\n   \\[\n   x + y = 10 + 0 = 10\n   \\]\n\n3. At \\( (0, 5) \\):\n   \\[\n   x + y = 0 + 5 = 5\n   \\]\n\n**Step 4: Identify the maximum value.**\n\nThe maximum value of \\( x + y \\) occurs at the point \\( (10, 0) \\) where \\( x + y = 10 \\).\n\n**Conclusion:**\n\nThe maximum value of \\( x + y \\) subject to the given constraints is **10** at the point **(10, 0)**.",
          494
        ],
        "time": 7.2479469776153564,
        "status": "success"
      },
      {
        "approach": "bon",
        "result": [
          "To maximize the objective function \\( z = x + y \\) subject to the constraints:\n\n1. \\( x + 2y \\leq 10 \\)\n2. \\( x \\geq 0 \\)\n3. \\( y \\geq 0 \\)\n\nwe can use the method of linear programming. \n\n### Step 1: Identify the constraints\nWe will first express the constraints and find the feasible region.\n\nFrom the constraint \\( x + 2y \\leq 10 \\), we can rewrite it as:\n\\[ 2y \\leq 10 - x \\quad \\Rightarrow \\quad y \\leq \\frac{10 - x}{2} \\]\n\nNow we sketch the line \\( x + 2y = 10 \\) to find its intersection points with the axes:\n\n- When \\( x = 0 \\):\n  \\[\n  0 + 2y = 10 \\quad \\Rightarrow \\quad y = 5\n  \\]\n  So the intersection with the y-axis is \\( (0, 5) \\).\n\n- When \\( y = 0 \\):\n  \\[\n  x + 0 = 10 \\quad \\Rightarrow \\quad x = 10\n  \\]\n  So the intersection with the x-axis is \\( (10, 0) \\).\n\n### Step 2: Determine the vertices of the feasible region\nThe feasible region is bound by the axes and the line \\( x + 2y = 10 \\).\n\nThe vertices of the feasible region are:\n- \\( (0, 0) \\)\n- \\( (0, 5) \\)\n- \\( (10, 0) \\)\n\n### Step 3: Evaluate the objective function at the vertices\nNow we evaluate \\( z = x + y \\) at the vertices:\n\n1. At \\( (0, 0) \\):\n   \\[\n   z = 0 + 0 = 0\n   \\]\n\n2. At \\( (0, 5) \\):\n   \\[\n   z = 0 + 5 = 5\n   \\]\n\n3. At \\( (10, 0) \\):\n   \\[\n   z = 10 + 0 = 10\n   \\]\n\n### Step 4: Find the maximum value\nThe maximum value of \\( z \\) occurs at the vertex \\( (10, 0) \\), where \\( z = 10 \\).\n\n### Conclusion\nThe maximum value of \\( z = x + y \\) subject to the given constraints is \\( 10 \\), obtained at the point \\( (10, 0) \\).",
          1719
        ],
        "time": 11.04929804801941,
        "status": "success"
      },
      {
        "approach": "plansearch",
        "result": [
          [
            "```python\nfrom scipy.optimize import linprog\n\n# Define the coefficients of the objective function (to maximize x + y, we minimize -x - y)\nc = [-1, -1]\n\n# Define the coefficients of the inequality constraints (Ax <= b)\n# x + 2y <= 10\nA = [[1, 2]]\nb = [10]\n\n# Define bounds for x and y (x >= 0, y >= 0)\nx_bounds = (0, None)\ny_bounds = (0, None)\n\n# Solve the linear programming problem\nresult = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method='highs')\n\n# The maximum value will be the negative of the result since we minimized -x - y\nmax_value = -result.fun\n\nprint(max_value)\n```"
          ],
          1831
        ],
        "time": 19.127686977386475,
        "status": "success"
      },
      {
        "approach": "moa",
        "result": [
          "To maximize the objective function \\( z = x + y \\) subject to the constraints:\n\n1. \\( x + 2y \\leq 10 \\)\n2. \\( x \\geq 0 \\)\n3. \\( y \\geq 0 \\)\n\nwe can follow these steps:\n\n### Step 1: Identify the Constraints\n\nFirst, we rewrite the constraint \\( x + 2y \\leq 10 \\) to find the intercepts:\n\n- **x-intercept** (set \\( y = 0 \\)):\n  \\[\n  x + 2(0) = 10 \\implies x = 10 \\quad \\Rightarrow \\quad (10, 0)\n  \\]\n- **y-intercept** (set \\( x = 0 \\)):\n  \\[\n  0 + 2y = 10 \\implies y = 5 \\quad \\Rightarrow \\quad (0, 5)\n  \\]\n\n### Step 2: Graph the Constraints\n\nThe line \\( x + 2y = 10 \\) intersects the axes at points \\( (10, 0) \\) and \\( (0, 5) \\). The feasible region defined by \\( x + 2y \\leq 10 \\) is below this line, along with the restrictions \\( x \\geq 0 \\) and \\( y \\geq 0 \\). This forms a triangular feasible region with vertices at:\n\n1. \\( (0, 0) \\)\n2. \\( (10, 0) \\)\n3. \\( (0, 5) \\)\n\n### Step 3: Evaluate the Objective Function at Each Vertex\n\nNow we evaluate \\( z = x + y \\) at each vertex:\n\n1. At \\( (0, 0) \\):\n   \\[\n   z = 0 + 0 = 0\n   \\]\n2. At \\( (10, 0) \\):\n   \\[\n   z = 10 + 0 = 10\n   \\]\n3. At \\( (0, 5) \\):\n   \\[\n   z = 0 + 5 = 5\n   \\]\n\n### Step 4: Determine the Maximum Value\n\nComparing the values of \\( z \\) at the corner points:\n- At \\( (0, 0) \\), \\( z = 0 \\)\n- At \\( (10, 0) \\), \\( z = 10 \\)\n- At \\( (0, 5) \\), \\( z = 5 \\)\n\nThe maximum value of \\( z \\) is \\( 10 \\) at the point \\( (10, 0) \\).\n\n### Conclusion\n\nThus, the maximum value of \\( x + y \\) subject to the given constraints is \\( \\boxed{10} \\), occurring at the point \\( (10, 0) \\).",
          2947
        ],
        "time": 20.96970796585083,
        "status": "success"
      },
      {
        "approach": "rto",
        "result": [
          "```markdown\nTo maximize the objective function \\( z = x + y \\) subject to the constraints:\n\n1. \\( x + 2y \\leq 10 \\)\n2. \\( x \\geq 0 \\)\n3. \\( y \\geq 0 \\)\n\n### Step 1: Define the Objective Function\nThe objective function we want to maximize is:\n\\[\nz = x + y\n\\]\n\n### Step 2: Define the Constraints\nThe constraints are:\n1. \\( x + 2y \\leq 10 \\)\n2. \\( x \\geq 0 \\)\n3. \\( y \\geq 0 \\)\n\n### Step 3: Graph the Constraints\nTo graph the constraint \\( x + 2y = 10 \\):\n- When \\( x = 0 \\): \\( 0 + 2y = 10 \\) \u2192 \\( y = 5 \\) (Point: \\( (0, 5) \\))\n- When \\( y = 0 \\): \\( x + 0 = 10 \\) \u2192 \\( x = 10 \\) (Point: \\( (10, 0) \\))\n\n### Step 4: Identify the Feasible Region\nThe feasible region is bounded by:\n- The line \\( x + 2y = 10 \\)\n- The x-axis (where \\( y = 0 \\))\n- The y-axis (where \\( x = 0 \\))\n\nThis region is in the first quadrant where both \\( x \\) and \\( y \\) are non-negative.\n\n### Step 5: Find Corner Points\nThe corner points of the feasible region are:\n1. \\( (0, 0) \\)\n2. \\( (0, 5) \\)\n3. \\( (10, 0) \\)\n\n### Step 6: Evaluate the Objective Function\nNow, we evaluate \\( z = x + y \\) at each corner point:\n1. At \\( (0, 0) \\): \\( z = 0 + 0 = 0 \\)\n2. At \\( (0, 5) \\): \\( z = 0 + 5 = 5 \\)\n3. At \\( (10, 0) \\): \\( z = 10 + 0 = 10 \\)\n\n### Step 7: Determine the Maximum Value\nComparing the values of \\( z \\):\n- At \\( (0, 0) \\): \\( z = 0 \\)\n- At \\( (0, 5) \\): \\( z = 5 \\)\n- At \\( (10, 0) \\): \\( z = 10 \\)\n\nThe maximum value of \\( z \\) is \\( 10 \\) at the point \\( (10, 0) \\).\n\n### Conclusion\nThe maximum value of the objective function \\( z = x + y \\) subject to the given constraints is \\( \\boxed{10} \\), occurring at the point \\( (10, 0) \\).\n```",
          2050
        ],
        "time": 25.46726679801941,
        "status": "success"
      },
      {
        "approach": "self_consistency",
        "result": [
          "To maximize the objective function \\( z = x + y \\) subject to the constraints:\n\n1. \\( x + 2y \\leq 10 \\)\n2. \\( x \\geq 0 \\)\n3. \\( y \\geq 0 \\)\n\nwe can use the method of linear programming.\n\n### Step 1: Graph the Constraints\n\n1. **Constraint 1:** Rearranging \\( x + 2y \\leq 10 \\)\n   - To find the intercepts:\n     - Set \\( y = 0 \\): \\( x + 2(0) = 10 \\)  \u2192 \\( x = 10 \\) (point (10, 0))\n     - Set \\( x = 0 \\): \\( 0 + 2y = 10 \\) \u2192 \\( y = 5 \\) (point (0, 5))\n   - This line can be represented as \\( x + 2y = 10 \\).\n\n2. **Constraints 2 and 3:** \\( x \\geq 0 \\) and \\( y \\geq 0 \\) imply that we are only interested in the first quadrant.\n\n### Step 2: Identify the Feasible Region\n\nThe feasible region is bounded by the axes and the line \\( x + 2y = 10 \\). \n\n### Step 3: Find the Vertices of the Feasible Region\n\nThe vertices of the feasible region occur at the intersections of the lines:\n\n1. Intersection of \\( x + 2y = 10 \\) and \\( x = 0 \\):\n   - Substituting \\( x = 0 \\) gives \\( 0 + 2y = 10 \\) \u2192 \\( y = 5 \\) (point (0, 5))\n\n2. Intersection of \\( x + 2y = 10 \\) and \\( y = 0 \\):\n   - Substituting \\( y = 0 \\) gives \\( x + 0 = 10 \\) \u2192 \\( x = 10 \\) (point (10, 0))\n\n3. The point where \\( x = 0 \\) and \\( y = 0 \\) is (0, 0).\n\n### Step 4: Evaluate the Objective Function at Each Vertex\n\nWe will evaluate \\( z = x + y \\) at each vertex:\n\n1. At point (0, 0):\n   \\[\n   z = 0 + 0 = 0\n   \\]\n\n2. At point (0, 5):\n   \\[\n   z = 0 + 5 = 5\n   \\]\n\n3. At point (10, 0):\n   \\[\n   z = 10 + 0 = 10\n   \\]\n\n### Step 5: Identify the Maximum Value\n\nThe maximum value of \\( z \\) occurs at the point (10, 0) where \\( z = 10 \\).\n\n### Conclusion\n\nThus, the maximum value of \\( x + y \\) subject to the given constraints is \\( \\boxed{10} \\), attained at the point \\( (10, 0) \\).",
          3030
        ],
        "time": 38.01216006278992,
        "status": "success"
      },
      {
        "approach": "rstar",
        "result": [
          "10",
          10488
        ],
        "time": 48.4237060546875,
        "status": "success"
      },
      {
        "approach": "mcts",
        "result": [
          "To maximize the objective function \\( z = x + y \\) subject to the constraints \\( x + 2y \\leq 10 \\), \\( x \\geq 0 \\), and \\( y \\geq 0 \\), we can follow these steps:\n\n1. **Identify the feasible region**:\n   - The constraint \\( x + 2y \\leq 10 \\) can be rearranged into the equation of a line:\n     \\[\n     y = \\frac{10 - x}{2}\n     \\]\n   - The line intersects the axes:\n     - When \\( x = 0 \\), \\( y = 5 \\) (point (0, 5)).\n     - When \\( y = 0 \\), \\( x = 10 \\) (point (10, 0)).\n   - The other constraints ensure we only consider the first quadrant where \\( x \\geq 0 \\) and \\( y \\geq 0 \\).\n   - Thus, the vertices of the feasible region are (0, 0), (10, 0), and (0, 5).\n\n2. **Evaluate the objective function at the vertices**:\n   - At (0, 0): \\( z = 0 + 0 = 0 \\)\n   - At (10, 0): \\( z = 10 + 0 = 10 \\)\n   - At (0, 5): \\( z = 0 + 5 = 5 \\)\n\n3. **Determine the maximum value**:\n   - Among the evaluated points, the maximum value of \\( z \\) occurs at (10, 0) where \\( z = 10 \\).\n\n**Conclusion**:\nThe maximum value of \\( x + y \\) subject to the given constraints is \\( \\boxed{10} \\), occurring at the point \\( (10, 0) \\).",
          9365
        ],
        "time": 64.1895182132721,
        "status": "success"
      },
      {
        "approach": "pvg",
        "result": [
          "To maximize the objective function \\( z = x + y \\) subject to the given constraints, we will follow these steps:\n\n1. **Identify the constraints**:\n   - \\( x + 2y \\leq 10 \\)\n   - \\( x \\geq 0 \\)\n   - \\( y \\geq 0 \\)\n\n2. **Graph the constraints**:\n   - We will first rewrite the inequality \\( x + 2y \\leq 10 \\) as an equation to find the boundary line:\n     \\[\n     x + 2y = 10\n     \\]\n     Rearranging gives:\n     \\[\n     2y = 10 - x \\quad \\Rightarrow \\quad y = \\frac{10 - x}{2}\n     \\]\n\n   - The x-intercept occurs when \\( y = 0 \\):\n     \\[\n     x + 2(0) = 10 \\quad \\Rightarrow \\quad x = 10 \\quad \\text{(point (10, 0))}\n     \\]\n\n   - The y-intercept occurs when \\( x = 0 \\):\n     \\[\n     0 + 2y = 10 \\quad \\Rightarrow \\quad y = 5 \\quad \\text{(point (0, 5))}\n     \\]\n\n   - Therefore, the line \\( x + 2y = 10 \\) intersects the axes at the points (10, 0) and (0, 5).\n\n3. **Plot the feasible region**:\n   - The feasible region is defined by the constraints \\( x \\geq 0 \\) and \\( y \\geq 0 \\), along with the area below the line \\( x + 2y = 10 \\) in the first quadrant.\n\n4. **Identify the corner points of the feasible region**:\n   - The corner points of the feasible region are:\n     - (0, 0) (the origin)\n     - (10, 0) (intercepts x-axis)\n     - (0, 5) (intercepts y-axis)\n\n5. **Evaluate the objective function at each corner point**:\n   - At (0, 0):\n     \\[\n     z = 0 + 0 = 0\n     \\]\n   - At (10, 0):\n     \\[\n     z = 10 + 0 = 10\n     \\]\n   - At (0, 5):\n     \\[\n     z = 0 + 5 = 5\n     \\]\n\n6. **Determine the maximum value**:\n   - The maximum value of \\( z \\) occurs at the point (10, 0) where \\( z = 10 \\).\n\n7. **Conclusion**:\n   - The maximum value of \\( x + y \\) subject to the given constraints is \\( \\boxed{10} \\), achieved at the point \\( (10, 0) \\).",
          39374
        ],
        "time": 84.64891982078552,
        "status": "success"
      }
    ]
  },
  {
    "test_case": {
      "name": "GSM8K",
      "system_prompt": "",
      "query": "If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot?"
    },
    "results": [
      {
        "approach": "re2",
        "result": [
          "If there are 3 cars in the parking lot and 2 more cars arrive, you would add the two amounts together: \n\n3 cars + 2 cars = 5 cars.\n\nSo, there would be 5 cars in the parking lot.",
          50
        ],
        "time": 0.8612151145935059,
        "status": "success"
      },
      {
        "approach": "leap",
        "result": [
          "To find the total number of cars in the parking lot, you can add the number of cars initially present to the number of cars that arrive. \n\nStarting with 3 cars and adding 2 more:\n\n3 (initial cars) + 2 (arriving cars) = 5 cars\n\nSo, there are 5 cars in the parking lot.",
          78
        ],
        "time": 1.8087499141693115,
        "status": "success"
      },
      {
        "approach": "cot_reflection",
        "result": [
          "There are 5 cars in the parking lot.",
          128
        ],
        "time": 2.0356550216674805,
        "status": "success"
      },
      {
        "approach": "bon",
        "result": [
          "To find the total number of cars in the parking lot, you would add the number of cars already there to the number of cars that arrive. \n\nStarting with 3 cars and adding 2 more:\n\n3 cars + 2 cars = 5 cars\n\nSo, there are 5 cars in the parking lot.",
          184
        ],
        "time": 3.007913112640381,
        "status": "success"
      },
      {
        "approach": "self_consistency",
        "result": [
          "If there are 3 cars in the parking lot and 2 more cars arrive, you simply add the two amounts together: \n\n3 cars + 2 cars = 5 cars. \n\nTherefore, there are 5 cars in the parking lot.",
          292
        ],
        "time": 4.503175735473633,
        "status": "success"
      },
      {
        "approach": "moa",
        "result": [
          "If there are 3 cars in the parking lot and 2 more cars arrive, you add the two amounts together: \n\n3 cars + 2 cars = 5 cars.\n\nTherefore, there are 5 cars in the parking lot.",
          621
        ],
        "time": 5.820193767547607,
        "status": "success"
      },
      {
        "approach": "z3",
        "result": [
          "There are a total of 5 cars in the parking lot after 2 more cars arrive.",
          19
        ],
        "time": 6.089645147323608,
        "status": "success"
      },
      {
        "approach": "rto",
        "result": [
          "```python\n# Initialize the number of cars in the parking lot\ntotal_cars = 3 + 2\n\n# Output the total number of cars\nprint(\"Total number of cars in the parking lot:\", total_cars)\n```",
          427
        ],
        "time": 8.253803014755249,
        "status": "success"
      },
      {
        "approach": "mcts",
        "result": [
          "To find the total number of cars in the parking lot, you would add the cars already there to the cars that arrive. \n\nStarting with 3 cars and adding 2 more:\n\n3 (existing cars) + 2 (arriving cars) = 5 cars in total.\n\nSo, there are 5 cars in the parking lot.",
          1036
        ],
        "time": 13.102774858474731,
        "status": "success"
      },
      {
        "approach": "rstar",
        "result": [
          "5",
          3781
        ],
        "time": 17.557404041290283,
        "status": "success"
      },
      {
        "approach": "plansearch",
        "result": [
          [
            "```python\n# Initial count of cars in the parking lot\ninitial_cars = 3\n\n# Number of cars that arrive\narriving_cars = 2\n\n# Calculate the total number of cars in the parking lot\ntotal_cars = initial_cars + arriving_cars\n\n# Output the total number of cars\nprint(total_cars)\n```"
          ],
          1121
        ],
        "time": 22.444913148880005,
        "status": "success"
      },
      {
        "approach": "pvg",
        "result": [
          "To determine how many cars are in the parking lot, we can follow these simple steps:\n\n1. **Identify the initial number of cars**: There are 3 cars already in the parking lot.\n\n2. **Identify the number of cars that arrive**: An additional 2 cars arrive.\n\n3. **Add the two quantities together**: We need to add the initial number of cars to the number of cars that arrived.\n\n   \\[\n   \\text{Total cars} = \\text{Initial cars} + \\text{Arriving cars} = 3 + 2\n   \\]\n\n4. **Calculate the sum**: \n\n   \\[\n   3 + 2 = 5\n   \\]\n\nTherefore, the total number of cars in the parking lot after the two additional cars arrive is **5 cars**.",
          45481
        ],
        "time": 53.22097301483154,
        "status": "success"
      }
    ]
  },
  {
    "test_case": {
      "name": "r/LocalLLaMA",
      "system_prompt": "",
      "query": "I have a dish of potatoes. The following statements are true: No potatoes of mine, that are new, have >been boiled. All my potatoes in this dish are fit to eat. No unboiled potatoes of mine are fit to eat. Are there any new potatoes in this dish?"
    },
    "results": [
      {
        "approach": "re2",
        "result": [
          "Let's analyze the statements step by step:\n\n1. **No potatoes of mine that are new have been boiled.** \n   - This means that if a potato is new, it cannot have been boiled.\n\n2. **All my potatoes in this dish are fit to eat.**\n   - This indicates that every potato in the dish is suitable for consumption.\n\n3. **No unboiled potatoes of mine are fit to eat.**\n   - This implies that if a potato has not been boiled, it is not fit to eat.\n\nGiven these statements, we can draw the following conclusions:\n\n- Since all the potatoes in the dish are fit to eat (statement 2), they must either be boiled or, if they are new, then they are also boiled.\n- However, since no new potatoes have been boiled (statement 1), this means that any new potato in this dish would have to be unboiled.\n- According to statement 3, unboiled potatoes are not fit to eat, which contradicts the second statement where all potatoes are said to be fit to eat.\n\nTherefore, since having any new potatoes in the dish would lead to a contradiction with the conditions set, we can conclude that **there are no new potatoes in this dish**.",
          254
        ],
        "time": 2.6748926639556885,
        "status": "success"
      },
      {
        "approach": "leap",
        "result": [
          "Based on the statements provided:\n\n1. No new potatoes have been boiled.\n2. All potatoes in the dish are fit to eat.\n3. No unboiled potatoes are fit to eat.\n\nFrom these points, we can infer the following:\n\n- Since all the potatoes in the dish are fit to eat (statement 2) and no unboiled potatoes are fit to eat (statement 3), it follows that all the potatoes in the dish must be boiled.\n- Since no new potatoes have been boiled (statement 1), there cannot be any new (unboiled) potatoes in the dish.\n\nTherefore, the answer to the query is: No, there are no new potatoes in this dish.",
          149
        ],
        "time": 2.9487528800964355,
        "status": "success"
      },
      {
        "approach": "cot_reflection",
        "result": [
          "No, there are no new potatoes in this dish.",
          308
        ],
        "time": 3.455225944519043,
        "status": "success"
      },
      {
        "approach": "bon",
        "result": [
          "Let's analyze the statements:\n\n1. No potatoes of mine that are new have been boiled.\n2. All my potatoes in this dish are fit to eat.\n3. No unboiled potatoes of mine are fit to eat.\n\nFrom statement 1, we can conclude that new potatoes cannot be boiled.\n\nFrom statement 3, we know that if there are any unboiled potatoes, they cannot be fit to eat.\n\nSince statement 2 states that all potatoes in the dish are fit to eat, it implies there cannot be any unboiled potatoes in the dish, because if there were, they would not be fit to eat per statement 3.\n\nGiven that:\n- If there are no unboiled potatoes, then all potatoes in the dish must be boiled (because all must be fit to eat).\n- Since new potatoes can't be boiled (from statement 1), it follows that there cannot be any new potatoes in the dish.\n\nTherefore, the conclusion is that there are no new potatoes in the dish.",
          680
        ],
        "time": 4.60465407371521,
        "status": "success"
      },
      {
        "approach": "z3",
        "result": [
          "Based on the analysis and the solver result, there are **no new potatoes in the dish**.",
          20
        ],
        "time": 10.212258100509644,
        "status": "success"
      },
      {
        "approach": "moa",
        "result": [
          "To determine if there are any new potatoes in the dish, let's analyze the statements provided:\n\n1. **No potatoes of mine that are new have been boiled.**  \n   This means that all new potatoes are unboiled.\n\n2. **All my potatoes in this dish are fit to eat.**  \n   This indicates that every potato in the dish is suitable for consumption.\n\n3. **No unboiled potatoes of mine are fit to eat.**  \n   This implies that only boiled potatoes can be considered fit to eat.\n\nNow, let's combine these statements:\n\n- If there are any new potatoes in the dish, they would be unboiled (from statement 1).\n- According to statement 3, unboiled potatoes are not fit to eat.\n- However, statement 2 asserts that all potatoes in the dish are fit to eat.\n\nThis creates a contradiction: if new potatoes were present, they would be unboiled and thus not fit to eat, which contradicts the assertion that all potatoes in the dish are fit to eat.\n\nTherefore, we can conclude that **there are no new potatoes in this dish.**",
          1278
        ],
        "time": 11.035154342651367,
        "status": "success"
      },
      {
        "approach": "rto",
        "result": [
          "```python\ndef analyze_potatoes():\n    # Define the statements\n    new_potatoes_boiled = False  # No new potatoes have been boiled\n    all_fit_to_eat = True         # All potatoes in the dish are fit to eat\n    unboiled_not_fit = True       # Unboiled potatoes are not fit to eat\n\n    # Analyze the implications\n    if all_fit_to_eat:\n        # If all potatoes are fit to eat, they must be boiled\n        all_boiled = True\n    else:\n        all_boiled = False\n\n    # New potatoes are unboiled and therefore not fit to eat\n    if new_potatoes_boiled == False and all_boiled:\n        # If all potatoes are boiled, then there cannot be any new potatoes\n        return False  # No new potatoes in the dish\n\n    return True  # New potatoes are present (this case won't occur based on the logic)\n\n# Final conclusion\nnew_potatoes_present = analyze_potatoes()\nprint(\"Are there any new potatoes in this dish?\", not new_potatoes_present)\n```",
          987
        ],
        "time": 11.263190984725952,
        "status": "success"
      },
      {
        "approach": "self_consistency",
        "result": [
          "Based on the statements provided, we can analyze the situation as follows:\n\n1. \"No potatoes of mine, that are new, have been boiled.\" This means that if a potato is new, it cannot be boiled.\n\n2. \"All my potatoes in this dish are fit to eat.\" This indicates that every potato in the dish is suitable for consumption.\n\n3. \"No unboiled potatoes of mine are fit to eat.\" This tells us that if a potato has not been boiled, it is not fit to eat.\n\nGiven these statements, we can draw the following conclusions:\n\n- For a potato to be fit to eat, it must be boiled. \n- Since all the potatoes in the dish are fit to eat, they must all be boiled.\n- Since no new potatoes can have been boiled (from the first statement), this eliminates the possibility of any new potatoes being present in the dish.\n\nTherefore, there cannot be any new potatoes in the dish. The answer is: **No, there are no new potatoes in this dish.**",
          1113
        ],
        "time": 11.989364862442017,
        "status": "success"
      },
      {
        "approach": "plansearch",
        "result": [
          [
            "```python\ndef has_new_potatoes():\n    # Analyzing the given observations logically\n    all_fit_to_eat = True  # All potatoes in the dish are fit to eat\n    no_unboiled_fit_to_eat = True  # No unboiled potatoes are fit to eat\n    no_new_boiled = True  # No new potatoes have been boiled\n\n    # Since all potatoes are fit to eat, they must all be boiled\n    if all_fit_to_eat and no_unboiled_fit_to_eat:\n        all_boiled = True\n    else:\n        all_boiled = False\n\n    # New potatoes cannot be boiled\n    if all_boiled and no_new_boiled:\n        return False  # There are no new potatoes in the dish\n\n    return True  # This case won't be hit based on our logic\n\n# Run the function to check for new potatoes\nnew_potatoes_present = has_new_potatoes()\nprint(f\"Are there any new potatoes in this dish? {'Yes' if new_potatoes_present else 'No'}\")\n```"
          ],
          1498
        ],
        "time": 16.858115196228027,
        "status": "success"
      },
      {
        "approach": "mcts",
        "result": [
          "Let's analyze the statements you provided:\n\n1. No potatoes of mine, that are new, have been boiled.\n2. All my potatoes in this dish are fit to eat.\n3. No unboiled potatoes of mine are fit to eat.\n\nFrom these statements, we can draw some conclusions:\n\n- From statement 3, if no unboiled potatoes are fit to eat, then all potatoes that are fit to eat must be boiled.\n- Statement 2 tells us that all potatoes in the dish are fit to eat, thus all must be boiled.\n- Statement 1 indicates that new potatoes have not been boiled.\n\nSince all the potatoes in your dish must be boiled (to be fit to eat according to statements 2 and 3), and no new potatoes have been boiled (according to statement 1), it follows that there cannot be any new potatoes in the dish.\n\nTherefore, the answer is: No, there are no new potatoes in this dish.",
          4515
        ],
        "time": 23.75214385986328,
        "status": "success"
      },
      {
        "approach": "rstar",
        "result": [
          "0",
          5843
        ],
        "time": 35.2991418838501,
        "status": "success"
      },
      {
        "approach": "pvg",
        "result": [
          "To determine if there are any new potatoes in your dish based on the statements provided, let's analyze them one by one:\n\n1. **No potatoes of mine, that are new, have been boiled.**\n   - This means that if a potato is new, it cannot be boiled. Therefore, all new potatoes remain unboiled.\n\n2. **All my potatoes in this dish are fit to eat.**\n   - This indicates that every potato in the dish is edible.\n\n3. **No unboiled potatoes of mine are fit to eat.**\n   - This means that if a potato has not been boiled, it cannot be fit to eat. \n\nNow, let's combine these statements:\n\n- From statement 1, we know that new potatoes are unboiled.\n- From statement 3, we know that no unboiled potatoes can be fit to eat.\n- However, according to statement 2, all potatoes in the dish are fit to eat.\n\nGiven that the new potatoes are unboiled (and thus not fit to eat), but all potatoes in the dish are fit to eat, we can conclude that there cannot be any new potatoes in the dish. If there were new potatoes, they would not meet the condition of being fit to eat, creating a contradiction.\n\nTherefore, the answer is:\n\n**No, there are no new potatoes in this dish.**",
          51999
        ],
        "time": 54.96053600311279,
        "status": "success"
      }
    ]
  },
  {
    "test_case": {
      "name": "reddit",
      "system_prompt": "",
      "query": "There are 24 volunteers. Over the next 3 weeks, each volunteer is assigned to a different task. There are 8 tasks. Each week, the volunteers switch tasks. Each task has 3 volunteers assigned to it. Volunteers cannot be assigned to the same task more than once, and volunteers cannot share the same task more than once."
    },
    "results": [
      {
        "approach": "re2",
        "result": [
          "To solve this problem, we need to assign the 24 volunteers to 8 tasks over the course of 3 weeks, with the following constraints:\n\n1. Each volunteer must be assigned to a different task each week.\n2. Each task has 3 volunteers assigned to it.\n3. No volunteer can be assigned to the same task more than once.\n4. No volunteers can share the same task more than once.\n\nGiven that there are 8 tasks, with 3 volunteers per task, the total number of volunteer spots available over 3 weeks is:\n\n- 8 tasks \u00d7 3 volunteers per task \u00d7 3 weeks = 72 volunteer-task assignments.\n\nSince we have 24 volunteers, each volunteer can be assigned to a different task in each of the 3 weeks, resulting in:\n\n- 24 volunteers \u00d7 3 tasks per volunteer = 72 total assignments (which matches the total number of spots available).\n\nNow, we need to design a schedule where each volunteer rotates through different tasks and does not repeat sharing or being assigned to the same task. Here's how we can approach it:\n\n**Week 1**:\n- Task 1: V1, V2, V3\n- Task 2: V4, V5, V6\n- Task 3: V7, V8, V9\n- Task 4: V10, V11, V12\n- Task 5: V13, V14, V15\n- Task 6: V16, V17, V18\n- Task 7: V19, V20, V21\n- Task 8: V22, V23, V24\n\n**Week 2** (Rotate the volunteers):\n- Task 1: V1, V4, V7\n- Task 2: V2, V5, V8\n- Task 3: V3, V6, V9\n- Task 4: V10, V13, V16\n- Task 5: V11, V14, V17\n- Task 6: V12, V15, V18\n- Task 7: V19, V22, V23\n- Task 8: V20, V21, V24\n\n**Week 3** (Another rotation):\n- Task 1: V1, V5, V9\n- Task 2: V2, V6, V10\n- Task 3: V3, V4, V11\n- Task 4: V12, V14, V17\n- Task 5: V13, V15, V18\n- Task 6: V19, V20, V22\n- Task 7: V21, V23, V24\n- Task 8: V7, V8, V16\n\nIn this way, each volunteer will work on a different task each week, and they do not repeat tasks or share tasks with the same group more than once.\n\nYou may have different rotations or assignments, but the important part of your planning is to ensure that the constraints are respected while providing equal distribution of tasks among volunteers.",
          645
        ],
        "time": 6.407229900360107,
        "status": "success"
      },
      {
        "approach": "leap",
        "result": [
          "To assign 24 volunteers to 8 tasks over the next 3 weeks under the given conditions, we can follow these steps:\n\n### Task Breakdown:\n- Each week, there are 8 tasks.\n- Each task has 3 volunteers assigned to it.\n- Therefore, each week there will be \\( 8 \\times 3 = 24 \\) volunteer assignments, which uses all 24 volunteers.\n\n### Weekly Schedule:\n1. **Week 1:** Assign volunteers to tasks. For example:\n   - Task 1: Volunteer 1, Volunteer 2, Volunteer 3\n   - Task 2: Volunteer 4, Volunteer 5, Volunteer 6\n   - Task 3: Volunteer 7, Volunteer 8, Volunteer 9\n   - Task 4: Volunteer 10, Volunteer 11, Volunteer 12\n   - Task 5: Volunteer 13, Volunteer 14, Volunteer 15\n   - Task 6: Volunteer 16, Volunteer 17, Volunteer 18\n   - Task 7: Volunteer 19, Volunteer 20, Volunteer 21\n   - Task 8: Volunteer 22, Volunteer 23, Volunteer 24\n\n2. **Week 2:** Volunteers switch tasks such that no one repeats a task with the same volunteers. This can be achieved by rotating the volunteers. For example:\n   - Task 1: Volunteer 1, Volunteer 4, Volunteer 7\n   - Task 2: Volunteer 2, Volunteer 5, Volunteer 8\n   - Task 3: Volunteer 3, Volunteer 6, Volunteer 9\n   - Task 4: Volunteer 10, Volunteer 13, Volunteer 16\n   - Task 5: Volunteer 11, Volunteer 14, Volunteer 17\n   - Task 6: Volunteer 12, Volunteer 15, Volunteer 18\n   - Task 7: Volunteer 19, Volunteer 22, Volunteer 24\n   - Task 8: Volunteer 20, Volunteer 21, Volunteer 23\n\n3. **Week 3:** Continue rotating volunteers with a new combination such that no one repeats a task with previous partners. For example:\n   - Task 1: Volunteer 1, Volunteer 10, Volunteer 19\n   - Task 2: Volunteer 2, Volunteer 11, Volunteer 20\n   - Task 3: Volunteer 3, Volunteer 12, Volunteer 21\n   - Task 4: Volunteer 4, Volunteer 13, Volunteer 22\n   - Task 5: Volunteer 5, Volunteer 14, Volunteer 23\n   - Task 6: Volunteer 6, Volunteer 15, Volunteer 24\n   - Task 7: Volunteer 7, Volunteer 16, Volunteer 17\n   - Task 8: Volunteer 8, Volunteer 9, Volunteer 18\n\n### Summary:\nThis way, over the 3 weeks, each volunteer gets to work on a different task while ensuring that no volunteer repeats a task with the same colleagues. The pattern of assignment rotates both tasks and volunteers consistently. Each volunteer will complete 3 different tasks over the three weeks while maintaining the conditions that no one shares a task more than once with other volunteers. \n\nThis method adheres to the requirement and offers a fair way to engage all volunteers across various tasks.",
          696
        ],
        "time": 6.751976013183594,
        "status": "success"
      },
      {
        "approach": "cot_reflection",
        "result": [
          "The task assignments for 24 volunteers over 3 weeks with 8 tasks can be structured so that each volunteer is assigned to a different task each week, ensuring no repeats and no shared tasks across the weeks. Each week, 3 volunteers will be assigned to each of the 8 tasks, ensuring compliance with the rules.",
          892
        ],
        "time": 10.223955154418945,
        "status": "success"
      },
      {
        "approach": "bon",
        "result": [
          "To structure the assignment of 24 volunteers to 8 tasks over 3 weeks, where each task has 3 volunteers, we need to ensure that:\n\n1. Each volunteer is assigned to a different task each week.\n2. No volunteer can share a task with the same volunteer more than once.\n3. Each task has exactly 3 volunteers at all times.\n\nWith this framework set up, here's one possible way to organize the tasks over the 3 weeks.\n\n### Week 1:\nLet's assign the volunteers (V1 to V24) to tasks (T1 to T8) as follows:\n\n- **Task 1 (T1)**: V1, V2, V3\n- **Task 2 (T2)**: V4, V5, V6\n- **Task 3 (T3)**: V7, V8, V9\n- **Task 4 (T4)**: V10, V11, V12\n- **Task 5 (T5)**: V13, V14, V15\n- **Task 6 (T6)**: V16, V17, V18\n- **Task 7 (T7)**: V19, V20, V21\n- **Task 8 (T8)**: V22, V23, V24\n\n### Week 2:\nNow we will rotate the volunteers in such a manner that no one is assigned to the same task or paired with the same volunteers. One way to do this is to shift the assignments:\n\n- **Task 1 (T1)**: V4, V5, V1\n- **Task 2 (T2)**: V6, V7, V2\n- **Task 3 (T3)**: V8, V9, V3\n- **Task 4 (T4)**: V10, V13, V12\n- **Task 5 (T5)**: V14, V15, V11\n- **Task 6 (T6)**: V16, V19, V18\n- **Task 7 (T7)**: V17, V20, V21\n- **Task 8 (T8)**: V22, V23, V24\n\n### Week 3:\nFinally, we complete the assignments with another rotation:\n\n- **Task 1 (T1)**: V7, V8, V10\n- **Task 2 (T2)**: V11, V14, V12\n- **Task 3 (T3)**: V13, V15, V16\n- **Task 4 (T4)**: V18, V19, V22\n- **Task 5 (T5)**: V20, V21, V24\n- **Task 6 (T6)**: V1, V2, V3\n- **Task 7 (T7)**: V4, V5, V6\n- **Task 8 (T8)**: V9, V17, V23\n\n### Summary:\nAfter 3 weeks, all volunteers have switched to different tasks each week, maintaining the necessary pairwise distinctions across the 3 weeks. The precise configuration of volunteers can be adjusted to fit any specific requirements or constraints, but the overall framework adheres to your limitations. \n\n(Note: The exact order of volunteers in the tasks can be refined to ensure that no volunteer is ever paired more than once with the same others across all weeks and tasks. This example is a simplified approach highlighting core scheduling principles rather than guaranteeing every specification is met in a single iteration.)",
          2230
        ],
        "time": 10.616811037063599,
        "status": "success"
      },
      {
        "approach": "z3",
        "result": [
          "To solve the problem of assigning 24 volunteers to 8 tasks over 3 weeks, we need to ensure that each volunteer is assigned to a different task each week, and that no volunteer is assigned to the same task more than once throughout the 3 weeks.\n\n### Breakdown of the Problem:\n\n1. **Total Volunteers**: 24\n2. **Total Tasks**: 8\n3. **Volunteers per Task**: 3\n4. **Weeks**: 3\n\n### Total Assignments:\n\n- Each week, there are 8 tasks, and each task has 3 volunteers assigned to it. Therefore, in one week, we have:\n  \\[\n  8 \\text{ tasks} \\times 3 \\text{ volunteers/task} = 24 \\text{ volunteers}\n  \\]\n- This means that all 24 volunteers can be assigned to tasks each week without any overlap.\n\n### Weekly Assignments:\n\n- Over 3 weeks, each volunteer will be assigned to a different task each week. Since there are 8 tasks and each volunteer can only be assigned to a task once, we need to ensure that the assignments are structured so that no volunteer repeats a task.\n\n### Possible Assignment Strategy:\n\n1. **Week 1**: Assign volunteers to tasks in a straightforward manner. For example:\n   - Task 1: Volunteers 1, 2, 3\n   - Task 2: Volunteers 4, 5, 6\n   - Task 3: Volunteers 7, 8, 9\n   - Task 4: Volunteers 10, 11, 12\n   - Task 5: Volunteers 13, 14, 15\n   - Task 6: Volunteers 16, 17, 18\n   - Task 7: Volunteers 19, 20, 21\n   - Task 8: Volunteers 22, 23, 24\n\n2. **Week 2**: Rotate the volunteers to different tasks. One way to do this is to shift the volunteers in a systematic way. For example:\n   - Task 1: Volunteers 4, 5, 6\n   - Task 2: Volunteers 7, 8, 9\n   - Task 3: Volunteers 10, 11, 12\n   - Task 4: Volunteers 13, 14, 15\n   - Task 5: Volunteers 16, 17, 18\n   - Task 6: Volunteers 19, 20, 21\n   - Task 7: Volunteers 22, 23, 24\n   - Task 8: Volunteers 1, 2, 3\n\n3. **Week 3**: Continue rotating the volunteers to ensure that no volunteer repeats a task. For example:\n   - Task 1: Volunteers 7, 8, 9\n   - Task 2: Volunteers 10, 11, 12\n   - Task 3: Volunteers 13, 14, 15\n   - Task 4: Volunteers 16, 17, 18\n   - Task 5: Volunteers 19, 20, 21\n   - Task 6: Volunteers 22, 23, 24\n   - Task 7: Volunteers 1, 2, 3\n   - Task 8: Volunteers 4, 5, 6\n\n### Summary:\n\nBy carefully rotating the volunteers each week, we can ensure that:\n- Each volunteer is assigned to a different task each week.\n- No volunteer is assigned to the same task more than once.\n- Each task has 3 volunteers assigned to it each week.\n\nThis systematic approach allows for a balanced and fair distribution of tasks among the volunteers over the 3-week period.",
          786
        ],
        "time": 19.02259874343872,
        "status": "success"
      },
      {
        "approach": "plansearch",
        "result": [
          [
            "```python\nimport itertools\n\ndef assign_volunteers_to_tasks(num_volunteers, num_tasks, num_weeks):\n    # Generate a list of volunteers\n    volunteers = list(range(1, num_volunteers + 1))\n\n    # Initialize assignment structure\n    assignment = {week: {task: [] for task in range(1, num_tasks + 1)} for week in range(1, num_weeks + 1)}\n\n    # Assign volunteers to tasks initially\n    for week in range(1, num_weeks + 1):\n        for task in range(1, num_tasks + 1):\n            task_assignments = []\n\n            # Calculate volunteers for this week and task\n            if week == 1:\n                # For the first week assign volunteers in blocks of 3\n                start_index = (task - 1) * 3\n                task_assignments = volunteers[start_index:start_index + 3]\n            else:\n                # For subsequent weeks use cyclic assignment\n                previous_assignments = assignment[week - 1][task]\n                prev_indices = [(volunteer - 1) for volunteer in previous_assignments]\n                rotated_indices = [(idx + 1) % num_volunteers for idx in prev_indices]\n                task_assignments = [volunteers[idx] for idx in rotated_indices]\n\n            # Ensure only unique volunteers are assigned to a task across weeks\n            while len(set(task_assignments)) < 3:\n                # Rotate the indices to avoid duplicates as needed\n                task_assignments = [task_assignments[-1]] + task_assignments[:-1]\n\n            assignment[week][task] = task_assignments\n\n    return assignment\n\ndef display_assignment(assignment):\n    for week, tasks in assignment.items():\n        print(f\"Week {week}:\")\n        for task, volunteers in tasks.items():\n            print(f\"  Task {task}: Volunteers {volunteers}\")\n\n# Assign 24 volunteers to 8 tasks over 3 weeks\nnum_volunteers = 24\nnum_tasks = 8\nnum_weeks = 3\n\nassignment = assign_volunteers_to_tasks(num_volunteers, num_tasks, num_weeks)\ndisplay_assignment(assignment)\n```"
          ],
          2031
        ],
        "time": 22.678685903549194,
        "status": "success"
      },
      {
        "approach": "moa",
        "result": [
          "To effectively assign 24 volunteers to 8 tasks over 3 weeks while adhering to the constraints that each volunteer must work on a different task each week, cannot repeat tasks, and cannot work with the same volunteers on the same task more than once, we can follow a structured approach. Here\u2019s a detailed breakdown of the assignment strategy:\n\n### Overview of the Problem\n- **Total Volunteers**: 24 (labeled V1 to V24)\n- **Total Tasks**: 8 (labeled T1 to T8)\n- **Volunteers per Task**: 3\n- **Total Assignments per Week**: 8 tasks \u00d7 3 volunteers/task = 24 volunteers\n\n### Constraints\n1. Each volunteer must work on a different task each week.\n2. No volunteer can repeat the same task over the 3 weeks.\n3. No pair of volunteers can work together on the same task more than once.\n\n### Week-by-Week Assignment\n\n#### Week 1:\nWe can start with a straightforward assignment:\n\n- **T1**: V1, V2, V3\n- **T2**: V4, V5, V6\n- **T3**: V7, V8, V9\n- **T4**: V10, V11, V12\n- **T5**: V13, V14, V15\n- **T6**: V16, V17, V18\n- **T7**: V19, V20, V21\n- **T8**: V22, V23, V24\n\n#### Week 2:\nFor the second week, we will rotate volunteers to ensure they do not repeat tasks or work with the same individuals. A systematic rotation can be applied:\n\n- **T1**: V1, V4, V7\n- **T2**: V2, V5, V8\n- **T3**: V3, V6, V9\n- **T4**: V10, V13, V16\n- **T5**: V11, V14, V17\n- **T6**: V12, V15, V18\n- **T7**: V19, V22, V20\n- **T8**: V21, V24, V23\n\n#### Week 3:\nFor the final week, we will again rotate volunteers to ensure compliance with all constraints:\n\n- **T1**: V2, V10, V19\n- **T2**: V3, V11, V20\n- **T3**: V1, V12, V21\n- **T4**: V4, V13, V22\n- **T5**: V5, V14, V23\n- **T6**: V6, V15, V24\n- **T7**: V7, V16, V18\n- **T8**: V8, V9, V17\n\n### Summary\nThis arrangement ensures that over the course of three weeks:\n- Each volunteer has a different task each week.\n- No volunteer repeats a task.\n- No pair of volunteers works together on the same task more than once.\n\n### Final Verification\nIt is essential to double-check that all constraints are met:\n- Each volunteer has worked on 3 different tasks.\n- Each task has had a unique combination of volunteers across the weeks.\n\nThis structured approach provides a clear and organized method for managing volunteer assignments while respecting all specified constraints. Adjustments can be made based on volunteer preferences or availability, but the core structure should remain intact to ensure fairness and compliance.",
          3631
        ],
        "time": 23.126257181167603,
        "status": "success"
      },
      {
        "approach": "rto",
        "result": [
          "```python\nimport random\n\n# Step 1: Initialize volunteers and assignments\nvolunteers = list(range(1, 25))  # Volunteers 1 to 24\nassignments = [[] for _ in range(3)]  # 3 weeks of assignments\ntask_assignments = {task: set() for task in range(8)}  # Track assigned volunteers for each task\n\ndef select_volunteers(volunteers, task):\n    # Get the volunteers who have not been assigned to this task in previous weeks\n    available_volunteers = [v for v in volunteers if v not in task_assignments[task]]\n    \n    # Randomly select 3 unique volunteers for the task\n    selected_volunteers = random.sample(available_volunteers, 3)\n    \n    # Update the task assignments to include the newly assigned volunteers\n    task_assignments[task].update(selected_volunteers)\n    \n    return selected_volunteers\n\n# Step 2: Assign volunteers to tasks\nfor week in range(3):\n    current_week_assignments = []\n    for task in range(8):\n        # Select 3 volunteers for the current task\n        selected_volunteers = select_volunteers(volunteers, task)\n        current_week_assignments.append(selected_volunteers)\n    assignments[week] = current_week_assignments\n\n# Step 3: Output the assignments\nfor week in range(3):\n    print(f\"Week {week + 1}:\")\n    for task in range(8):\n        print(f\"  Task {task + 1}: Volunteers {assignments[week][task]}\")\n```",
          2140
        ],
        "time": 26.735251903533936,
        "status": "success"
      },
      {
        "approach": "mcts",
        "result": [
          "Given that you have 24 volunteers and 8 different tasks, you are assigning volunteers in such a way that following the constraints:\n\n1. **Volunteers are assigned to different tasks each week.**\n2. **Each task has 3 volunteers.**\n3. **Volunteers cannot be assigned to the same task more than once.**\n4. **Volunteers cannot share the same task more than once.**\n\n### Assignment Structure\n\n- **Total Number of Assignments per Week**: Since there are 8 tasks and each task has 3 volunteers, there are a total of \\(8 \\text{ tasks} \\times 3 \\text{ volunteers/task} = 24 \\text{ volunteers}\\) needed every week, which perfectly fits the 24 volunteers.\n\n### Assignments Over 3 Weeks\n\n#### Week 1:\n1. **Task 1**: Volunteer 1, Volunteer 2, Volunteer 3\n2. **Task 2**: Volunteer 4, Volunteer 5, Volunteer 6\n3. **Task 3**: Volunteer 7, Volunteer 8, Volunteer 9\n4. **Task 4**: Volunteer 10, Volunteer 11, Volunteer 12\n5. **Task 5**: Volunteer 13, Volunteer 14, Volunteer 15\n6. **Task 6**: Volunteer 16, Volunteer 17, Volunteer 18\n7. **Task 7**: Volunteer 19, Volunteer 20, Volunteer 21\n8. **Task 8**: Volunteer 22, Volunteer 23, Volunteer 24\n\n#### Week 2:\n1. **Task 1**: Volunteer 4, Volunteer 7, Volunteer 10\n2. **Task 2**: Volunteer 1, Volunteer 8, Volunteer 11\n3. **Task 3**: Volunteer 2, Volunteer 5, Volunteer 13\n4. **Task 4**: Volunteer 3, Volunteer 6, Volunteer 14\n5. **Task 5**: Volunteer 9, Volunteer 12, Volunteer 15\n6. **Task 6**: Volunteer 16, Volunteer 19, Volunteer 22\n7. **Task 7**: Volunteer 17, Volunteer 20, Volunteer 23\n8. **Task 8**: Volunteer 18, Volunteer 21, Volunteer 24\n\n#### Week 3:\n1. **Task 1**: Volunteer 2, Volunteer 5, Volunteer 18\n2. **Task 2**: Volunteer 3, Volunteer 19, Volunteer 22\n3. **Task 3**: Volunteer 1, Volunteer 20, Volunteer 23\n4. **Task 4**: Volunteer 7, Volunteer 12, Volunteer 21\n5. **Task 5**: Volunteer 4, Volunteer 17, Volunteer 24\n6. **Task 6**: Volunteer 6, Volunteer 15, Volunteer 9\n7. **Task 7**: Volunteer 8, Volunteer 10, Volunteer 14\n8. **Task 8**: Volunteer 11, Volunteer 13, Volunteer 16\n\n### Validation\n- Every volunteer is assigned to a different task each week.\n- No volunteer is assigned to the same task more than once across the three weeks.\n- Each task continues to have 3 different volunteers assigned while rotating effectively.\n\n### Final Notes\nThis solution respects all constraints given in the problem statement. It can be adjusted in its specifics as long as the conditions are fully met, maintaining the unique assignments for each volunteer and task throughout the weeks.",
          8266
        ],
        "time": 45.051475048065186,
        "status": "success"
      },
      {
        "approach": "self_consistency",
        "result": [
          "Given the constraints you mentioned, we can lay out a schedule for the 24 volunteers over the course of 3 weeks, ensuring that each of the 8 tasks has 3 different volunteers assigned to it every week. \n\nLet's denote the volunteers as \\( V_1, V_2, \\ldots, V_{24} \\) and the tasks as \\( T_1, T_2, \\ldots, T_8 \\).\n\n### Week 1\n- **Task 1**: \\( V_1, V_2, V_3 \\)\n- **Task 2**: \\( V_4, V_5, V_6 \\)\n- **Task 3**: \\( V_7, V_8, V_9 \\)\n- **Task 4**: \\( V_{10}, V_{11}, V_{12} \\)\n- **Task 5**: \\( V_{13}, V_{14}, V_{15} \\)\n- **Task 6**: \\( V_{16}, V_{17}, V_{18} \\)\n- **Task 7**: \\( V_{19}, V_{20}, V_{21} \\)\n- **Task 8**: \\( V_{22}, V_{23}, V_{24} \\)\n\n### Week 2\nNow, we need to rotate the assignments, ensuring no volunteer is assigned to the same task they had in Week 1.\n\n- **Task 1**: \\( V_4, V_5, V_6 \\) (previously in Task 2)\n- **Task 2**: \\( V_7, V_8, V_9 \\) (previously in Task 3)\n- **Task 3**: \\( V_{10}, V_{11}, V_{12} \\) (previously in Task 4)\n- **Task 4**: \\( V_{13}, V_{14}, V_{15} \\) (previously in Task 5)\n- **Task 5**: \\( V_{16}, V_{17}, V_{18} \\) (previously in Task 6)\n- **Task 6**: \\( V_{19}, V_{20}, V_{21} \\) (previously in Task 7)\n- **Task 7**: \\( V_{22}, V_{23}, V_{24} \\) (previously in Task 8)\n- **Task 8**: \\( V_1, V_2, V_3 \\) (previously in Task 1)\n\n### Week 3\nAgain, we rotate the assignments avoiding any repetitions for the volunteers.\n\n- **Task 1**: \\( V_7, V_8, V_9 \\) (previously in Task 3)\n- **Task 2**: \\( V_{10}, V_{11}, V_{12} \\) (previously in Task 4)\n- **Task 3**: \\( V_{13}, V_{14}, V_{15} \\) (previously in Task 5)\n- **Task 4**: \\( V_{16}, V_{17}, V_{18} \\) (previously in Task 6)\n- **Task 5**: \\( V_{19}, V_{20}, V_{21} \\) (previously in Task 7)\n- **Task 6**: \\( V_{22}, V_{23}, V_{24} \\) (previously in Task 8)\n- **Task 7**: \\( V_1, V_2, V_3 \\) (previously in Task 1)\n- **Task 8**: \\( V_4, V_5, V_6 \\) (previously in Task 2)\n\nWith the above distribution, every week has 3 volunteers assigned to each task. Each volunteer is rotated such that they do not repeat their task assignments across the 3 weeks, and no volunteer shares a task with another volunteer more than once. This meets your requirement succinctly.",
          3559
        ],
        "time": 51.43106269836426,
        "status": "success"
      },
      {
        "approach": "rstar",
        "result": [
          "72",
          12276
        ],
        "time": 70.31652808189392,
        "status": "success"
      },
      {
        "approach": "pvg",
        "result": [
          "To tackle the problem of assigning 24 volunteers to 8 tasks over 3 weeks while meeting the specified conditions, we can use a systematic approach. Here\u2019s a step-by-step breakdown of the assignment process:\n\n### Step 1: Understanding the Requirements\n\n1. **Total Volunteers:** 24\n2. **Total Tasks:** 8\n3. **Weeks:** 3\n4. **Volunteers per Task per Week:** 3\n5. **Constraints:**\n   - Each volunteer must switch tasks weekly.\n   - No volunteer can repeat a task in the 3-week period.\n   - No two volunteers can share the same task more than once.\n\n### Step 2: Task Assignment Strategy\n\nTo ensure that the assignments are balanced and fair, we can use a round-robin approach combined with a matrix to visualize the assignments. We will create a 3x8 matrix (for 3 weeks and 8 tasks) where each cell contains the IDs of volunteers assigned to that task.\n\n#### Assignment Process\n\n1. **Matrix Setup:**\n   - Create a 3x8 matrix to represent the weeks and tasks.\n   - Each task in a week will have 3 volunteers assigned.\n\n2. **Initial Random Assignment:**\n   - Randomly assign volunteers to tasks for Week 1.\n   - Ensure that each task has exactly 3 unique volunteers.\n\n3. **Subsequent Assignments:**\n   - For Week 2, rotate the volunteers to different tasks ensuring:\n     - No volunteer is assigned to the same task as in Week 1.\n     - Tasks still have 3 volunteers each.\n   - For Week 3, repeat the rotation ensuring the same conditions.\n\n### Step 3: Visual Representation\n\nHere\u2019s a sample matrix representation of volunteer assignments:\n\n| Week/Task | Task 1 | Task 2 | Task 3 | Task 4 | Task 5 | Task 6 | Task 7 | Task 8 |\n|-----------|--------|--------|--------|--------|--------|--------|--------|--------|\n| Week 1    | 1,2,3  | 4,5,6  | 7,8,9  | 10,11,12 | 13,14,15 | 16,17,18 | 19,20,21 | 22,23,24 |\n| Week 2    | 1,4,7  | 2,5,8  | 3,6,9  | 10,13,16 | 11,14,17 | 12,15,18 | 19,22,23 | 20,21,24 |\n| Week 3    | 1,5,9  | 2,6,10 | 3,7,11 | 4,8,12  | 13,19,21 | 14,20,22 | 15,18,24 | 16,17,23 |\n\n### Step 4: Verification of Constraints\n\nTo validate that all constraints are satisfied:\n\n1. **Check Unique Assignments:**\n   - Ensure no volunteer is assigned to the same task in different weeks.\n   - Verify that no two volunteers share the same task more than once across weeks.\n\n2. **Task Capacity:**\n   - Confirm that each task has exactly 3 volunteers assigned in each week.\n\n### Step 5: Adjustments for Absences\n\nIf a volunteer is unable to participate in a week, we can make the following adjustments:\n\n1. **Identify the Absent Volunteer:**\n   - Determine which task they were assigned to that week.\n\n2. **Reassigning Tasks:**\n   - Select a volunteer who has not been assigned to that task in the current or previous weeks and assign them to fill the vacancy. Ensure that the new assignment follows the original constraints.\n\n3. **Revalidate:**\n   - After making the adjustments, recheck the matrix to ensure all constraints are still satisfied.\n\n### Conclusion\n\nThis systematic approach allows for fair and balanced task assignments for 24 volunteers over 3 weeks, ensuring that all conditions are met. The matrix provides a clear visual representation of the assignments, and the validation steps ensure compliance with the requirements. Adjustments can be made easily for absences without disrupting the overall structure.",
          66754
        ],
        "time": 100.83396935462952,
        "status": "success"
      }
    ]
  }
]