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_get_partial_codes now caches input_ids greedily whenever it's given new ones. The caller is responsible for maintaining the validity of the cache. In particular, is_valid has to remove the last input_id from the cache where the last input_id isn't valid.
When benchmarking on the following settings, we see an 80% reduction in the time spent in syncode.
"""
Print all primes between 1 and n
"""
'''