





'''simple docstring'''







from __future__ import annotations

from math import gcd





def 		snake_case_							(_a			:							int					,     _a			:							int = 2					,     _a			:							int = 1					,     _a			:							int = 3					,     ):
    # A value less than 2 can cause an infinite loop in the algorithm.
    if num < 2:
        raise ValueError('''The input value cannot be less than 2'''  )

    # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
    # algorithm struggles to find factors that are divisible by two.
    # As a workaround, we specifically check for two and even inputs.
    #   See: https://math.stackexchange.com/a/2856214/165820
    if num > 2 and num % 2 == 0:
        return 2

    # Pollard's Rho algorithm requires a function that returns pseudorandom
    # values between 0 <= X < ``num``.  It doesn't need to be random in the
    # sense that the output value is cryptographically secure or difficult
    # to calculate, it only needs to be random in the sense that all output
    # values should be equally likely to appear.
    # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
    # However, the success of Pollard's algorithm isn't guaranteed and is
    # determined in part by the initial seed and the chosen random function.
    # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
    # where ``C`` is a value that we can modify between each attempt.
    def rand_fn(_a			:							int					,     _a			:							int					,     _a			:							int  ) -> int:
        return (pow(_a					,     2  ) + step) % modulus

    for _ in range(_a  ):
        # These track the position within the cycle detection logic.
        UpperCAmelCase     =	seed
        UpperCAmelCase     =	seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            UpperCAmelCase     =	rand_fn(_a					,     _a					,     _a  )
            UpperCAmelCase     =	rand_fn(_a					,     _a					,     _a  )
            UpperCAmelCase     =	rand_fn(_a					,     _a					,     _a  )

            # At some point both the tortoise and the hare will enter a cycle whose
            # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
            # the tortoise and hare will end up on the same value modulo ``p``.
            # We can detect when this happens because the position difference between
            # the tortoise and the hare will share a common divisor with ``num``.
            UpperCAmelCase     =	gcd(hare - tortoise					,     _a  )

            if divisor == 1:
                # No common divisor yet, just keep searching.
                continue
            else:
                # We found a common divisor!
                if divisor == num:
                    # Unfortunately, the divisor is ``num`` itself and is useless.
                    break
                else:
                    # The divisor is a nontrivial factor of ``num``!
                    return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
        UpperCAmelCase     =	hare

        # The new step value for the random function can just be incremented.
        # At first the results will be similar to what the old function would
        # have produced, but the value will quickly diverge after a bit.
        step += 1

    # We haven't found a divisor within the requested number of attempts.
    # We were unlucky or ``num`` itself is actually prime.
    return None


if __name__ == "__main__":
   import argparse

   A									=argparse.ArgumentParser()
   parser.add_argument(
       'num',
       type=int,
       help='The value to find a divisor of',
   )
   parser.add_argument(
       '--attempts',
       type=int,
       default=3,
       help='The number of attempts before giving up',
   )
   A									=parser.parse_args()

   A									=pollard_rho(args.num, attempts=args.attempts)
   if divisor is None:
      print(f"""{args.num} is probably prime""")
   else:
      A									=args.num // divisor
      print(f"""{args.num} = {divisor} * {quotient}""")










'''simple docstring'''







from __future__ import annotations





def 		snake_case_							(_a			:							str					,     _a			:							list[str] | None = None  ):
    UpperCAmelCase     =	word_bank or []
    # create a table
    UpperCAmelCase     =	len(_a  ) + 1

    UpperCAmelCase     =	[]
    for _ in range(_a  ):
        table.append([]  )
    # seed value
    UpperCAmelCase     =	[[]]  # because empty string has empty combination

    # iterate through the indices
    for i in range(_a  ):
        # condition
        if table[i] != []:
            for word in word_bank:
                # slice condition
                if target[i : i + len(_a  )] == word:
                    UpperCAmelCase     =	[
                        [word, *way] for way in table[i]
                    ]
                    # adds the word to every combination the current position holds
                    # now,push that combination to the table[i+len(word)]
                    table[i + len(_a  )] += new_combinations

    # combinations are in reverse order so reverse for better output
    for combination in table[len(_a  )]:
        combination.reverse()

    return table[len(_a  )]


if __name__ == "__main__":
   print(all_construct('jwajalapa', ['jwa', 'j', 'w', 'a', 'la', 'lapa']))
   print(all_construct('rajamati', ['s', 'raj', 'amat', 'raja', 'ma', 'i', 't']))
   print(
       all_construct(
           'hexagonosaurus',
           ['h', 'ex', 'hex', 'ag', 'ago', 'ru', 'auru', 'rus', 'go', 'no', 'o', 's'],
       )
   )




