

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.
		_UpperCamelCase   :		Optional[Any]			=						seed
		_UpperCamelCase   :		Optional[int]			=						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   :		Any			=						rand_fn(UpperCAmelCase_							,       UpperCAmelCase_							,       UpperCAmelCase_	)
			_UpperCamelCase   :		str			=						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   :		Any			=						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   :		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

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

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







from .glue import glue_convert_examples_to_features, glue_output_modes, glue_processors, glue_tasks_num_labels
from .squad import SquadExample, SquadFeatures, SquadVaProcessor, SquadVaProcessor, squad_convert_examples_to_features
from .utils import DataProcessor, InputExample, InputFeatures, SingleSentenceClassificationProcessor
from .xnli import xnli_output_modes, xnli_processors, xnli_tasks_num_labels




