





'''simple docstring'''





import re

from filelock import FileLock


try:
  import nltk

  UpperCAmelCase_  							=				True
except (ImportError, ModuleNotFoundError):
  UpperCAmelCase_  							=				False

if NLTK_AVAILABLE:
  with FileLock('.lock') as lock:
    nltk.download('punkt', quiet=True)




def 	lowerCAmelCase_					(						__UpperCAmelCase:    str   )   ->			Optional[int]:
    re.sub('''<n>'''	,     ''''''	,     __lowerCAmelCase   )  # remove pegasus newline char
    assert NLTK_AVAILABLE, "nltk must be installed to separate newlines between sentences. (pip install nltk)"
    return "\n".join(nltk.sent_tokenize(__lowerCAmelCase   )   )








from __future__ import annotations

from math import gcd




def 	lowerCAmelCase_					(						__UpperCAmelCase:    int	,     __UpperCAmelCase:    int = 2	,     __UpperCAmelCase:    int = 1	,     __UpperCAmelCase:    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(__UpperCAmelCase:    int	,     __UpperCAmelCase:    int	,     __UpperCAmelCase:    int   ) -> int:
        return (pow(__UpperCAmelCase	,     2   ) + step) % modulus

    for _ in range(__UpperCAmelCase   ):
        # These track the position within the cycle detection logic.
        UpperCamelCase__							:       List[Any]      =   seed
        UpperCamelCase__							:       List[str]      =   seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            UpperCamelCase__							:       Optional[int]      =   rand_fn(__UpperCAmelCase	,     __UpperCAmelCase	,     __UpperCAmelCase   )
            UpperCamelCase__							:       List[str]      =   rand_fn(__UpperCAmelCase	,     __UpperCAmelCase	,     __UpperCAmelCase   )
            UpperCamelCase__							:       Optional[int]      =   rand_fn(__UpperCAmelCase	,     __UpperCAmelCase	,     __UpperCAmelCase   )

            # 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__							:       Optional[int]      =   gcd(hare - tortoise	,     __UpperCAmelCase   )

            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__							:       List[str]      =   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

  UpperCAmelCase_  							=				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',
  )
  UpperCAmelCase_  							=				parser.parse_args()

  UpperCAmelCase_  							=				pollard_rho(args.num, attempts=args.attempts)
  if divisor is None:
    print(F'''{args.num} is probably prime''')
  else:
    UpperCAmelCase_  							=				args.num // divisor
    print(F'''{args.num} = {divisor} * {quotient}''')

