




"""simple docstring"""




from __future__ import annotations

from typing import Any

class   UpperCAmelCase_    :
		def __init__(						self				,						UpperCamelCase_ = 6       )						->       None:
					__lowercase			:  Node | None								=       None
					__lowercase			:  Node | None								=       None
					self.create_linked_list(UpperCamelCase_       )
		def 	_lowerCamelCase							(						self				,						UpperCamelCase_       )						->       None:
					__lowercase			:  int								=       Node()
					__lowercase			:  Tuple								=       current_node
					__lowercase			:  Optional[Any]								=       current_node
					__lowercase			:  Union[str, Any]								=       current_node
					for _ in range(1				,						UpperCamelCase_       ):
								__lowercase			:  str								=       Node()
								__lowercase			:  Optional[Any]								=       current_node
								__lowercase			:  List[Any]								=       previous_node
								__lowercase			:  Union[str, Any]								=       current_node
					__lowercase			:  Any								=       self.front
					__lowercase			:  Optional[Any]								=       previous_node
		def 	_lowerCamelCase							(						self       )						->       bool:
					return (
					    self.front == self.rear
					    and self.front is not None
					    and self.front.data is None
					)
		def 	_lowerCamelCase							(						self       )						->       Any | None:
					self.check_can_perform_operation()
					return self.front.data if self.front else None
		def 	_lowerCamelCase							(						self				,						UpperCamelCase_       )						->       None:
					if self.rear is None:
								return

					self.check_is_full()
					if not self.is_empty():
								__lowercase			:  Dict								=       self.rear.next
					if self.rear:
								__lowercase			:  Optional[Any]								=       data
		def 	_lowerCamelCase							(						self       )						->       Any:
					self.check_can_perform_operation()
					if self.rear is None or self.front is None:
								return None
					if self.front == self.rear:
								__lowercase			:  Dict								=       self.front.data
								__lowercase			:  Tuple								=       None
								return data

					__lowercase			:  Optional[Any]								=       self.front
					__lowercase			:  Dict								=       old_front.next
					__lowercase			:  int								=       old_front.data
					__lowercase			:  Optional[Any]								=       None
					return data
		def 	_lowerCamelCase							(						self       )						->       None:
					if self.is_empty():
								raise Exception('''Empty Queue'''       )
		def 	_lowerCamelCase							(						self       )						->       None:
					if self.rear and self.rear.next == self.front:
								raise Exception('''Full Queue'''       )


class   UpperCAmelCase_    :
		def __init__(						self       )						->       None:
					__lowercase			:  Any | None								=       None
					__lowercase			:  Node | None								=       None
					__lowercase			:  Node | None								=       None


if __name__ == "__main__":
			import doctest

			doctest.testmod()






from __future__ import annotations

from math import gcd


def 							__lowerCAmelCase     (							_UpperCamelCase		:	int		,				_UpperCamelCase		:	int = 2		,				_UpperCamelCase		:	int = 1		,				_UpperCamelCase		:	int = 3		,				)		->				int | None:
 '''simple docstring'''


 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.
  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(_UpperCamelCase		,				_UpperCamelCase		,				_UpperCamelCase						)
   SCREAMING_SNAKE_CASE		    =      rand_fn(_UpperCamelCase		,				_UpperCamelCase		,				_UpperCamelCase						)
   SCREAMING_SNAKE_CASE		    =      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``.
   SCREAMING_SNAKE_CASE		    =      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.
  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

      a_     :    int     		=   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_     :    Tuple     		=   parser.parse_args()

      a_     :    int     		=   pollard_rho(args.num, attempts=args.attempts)
      if divisor is None:
            print(F"""{args.num} is probably prime""")
      else:
            a_     :    Union[str, Any]     		=   args.num // divisor
            print(F"""{args.num} = {divisor} * {quotient}""")

