





'''simple docstring'''

from __future__ import annotations

from math import gcd





def lowerCamelCase						(     __lowerCamelCase :    int					,       __lowerCamelCase :    int = 2					,       __lowerCamelCase :    int = 1					,       __lowerCamelCase :    int = 3					,       )						->int | None:
    # 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(__lowerCamelCase :    int					,       __lowerCamelCase :    int					,       __lowerCamelCase :    int    ) -> int:
        return (pow(__lowerCamelCase					,       2    ) + step) % modulus

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

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

            # 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``.
            _SCREAMING_SNAKE_CASE									=			gcd(hare - tortoise					,       __lowerCamelCase    )

            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.
        _SCREAMING_SNAKE_CASE									=			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

      lowercase_					= 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""",
      )
      lowercase_					= parser.parse_args()

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










'''simple docstring'''

from typing import List

from .keymap import KEYMAP, get_character





def lowerCamelCase						(     __lowerCamelCase :    str    )						->Optional[int]:

    def decorator(__lowerCamelCase :    int    ):
        _SCREAMING_SNAKE_CASE									=			getattr(__lowerCamelCase					,       """handle_key"""					,       []    )
        handle += [key]
        setattr(__lowerCamelCase					,       """handle_key"""					,       __lowerCamelCase    )
        return func

    return decorator





def lowerCamelCase						(     *__lowerCamelCase :    List[str]    )						->Dict:

    def decorator(__lowerCamelCase :    int    ):
        _SCREAMING_SNAKE_CASE									=			getattr(__lowerCamelCase					,       """handle_key"""					,       []    )
        handle += keys
        setattr(__lowerCamelCase					,       """handle_key"""					,       __lowerCamelCase    )
        return func

    return decorator


class 					a_     (				snake_case_       ):

    '''simple docstring'''


    def __new__(		cls ,		A ,		A ,		A       )		->   int:
        _SCREAMING_SNAKE_CASE									=			super().__new__(cls ,		A ,		A ,		A       )
        if not hasattr(A ,		"""key_handler"""       ):
            setattr(A ,		"""key_handler""" ,		{}       )
        setattr(A ,		"""handle_input""" ,		KeyHandler.handle_input       )

        for value in attrs.values():
            _SCREAMING_SNAKE_CASE									=			getattr(A ,		"""handle_key""" ,		[]       )
            for key in handled_keys:
                _SCREAMING_SNAKE_CASE									=			value
        return new_cls





    @staticmethod
    def 			snake_case_(		cls       )		->   str:
        _SCREAMING_SNAKE_CASE									=			get_character()
        if char != KEYMAP["undefined"]:
            _SCREAMING_SNAKE_CASE									=			ord(A       )
        _SCREAMING_SNAKE_CASE									=			cls.key_handler.get(A       )
        if handler:
            _SCREAMING_SNAKE_CASE									=			char
            return handler(cls       )
        else:
            return None





def lowerCamelCase						(     cls :    Any    )						->Dict:
    return KeyHandler(cls.__name__					,       cls.__bases__					,       cls.__dict__.copy()    )



