





"""simple docstring"""






from __future__ import annotations

from math import gcd

def       __A (     a_  :int				,    a_  :int = 2				,    a_  :int = 1				,    a_  :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(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.
        __a	:  Any         =					seed
        __a	:  int         =					seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            __a	:  Optional[Any]         =					rand_fn(a_				,    a_				,    a_)
            __a	:  int         =					rand_fn(a_				,    a_				,    a_)
            __a	:  List[Any]         =					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``.
            __a	:  int         =					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.
        __a	:  Optional[int]         =					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

import sys
from collections import deque
from typing import Generic, TypeVar

A	     =					TypeVar('''T''')




class 			__lowercase     (     Generic[T]							):







					'''simple docstring'''




					__lowerCAmelCase					=					42  # Cache store of keys
					__lowerCAmelCase					=					42  # References of the keys in cache
					__lowerCAmelCase					=					10  # Maximum capacity of cache







					def __init__(  self		,     _UpperCAmelCase       ):
									__a	:  Optional[int]         =					deque()
									__a	:  Dict         =					set()
									if not n:
													__a	:  List[Any]         =					sys.maxsize
									elif n < 0:
													raise ValueError('''n should be an integer greater than 0.'''       )
									else:
													__a	:  str         =					n







					def 	_lowerCamelCase						(  self		,     _UpperCAmelCase       ):
									if x not in self.key_reference:
													if len(self.dq_store       ) == LRUCache._MAX_CAPACITY:
																	__a	:  int         =					self.dq_store.pop()
																	self.key_reference.remove(_UpperCAmelCase       )
									else:
													self.dq_store.remove(_UpperCAmelCase       )

									self.dq_store.appendleft(_UpperCAmelCase       )
									self.key_reference.add(_UpperCAmelCase       )







					def 	_lowerCamelCase						(  self       ):
									for k in self.dq_store:
													print(_UpperCAmelCase       )







					def __repr__(  self       ):
									return f"""LRUCache({self._MAX_CAPACITY}) => {list(self.dq_store       )}"""


if __name__ == "__main__":
		import doctest

		doctest.testmod()

		A	     =					LRUCache(4)
		lru_cache.refer('''A''')
		lru_cache.refer(2)
		lru_cache.refer(3)
		lru_cache.refer('''A''')
		lru_cache.refer(4)
		lru_cache.refer(5)
		lru_cache.display()

		print(lru_cache)
		assert str(lru_cache) == "LRUCache(4) => [5, 4, 'A', 3]"