



from __future__ import annotations

from math import gcd






def      _A				(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.
						lowerCamelCase__ :      int								=		seed
						lowerCamelCase__ :      str								=		seed

						while True:
									# At each iteration, the tortoise moves one step and the hare moves two.
									lowerCamelCase__ :      List[Any]								=		rand_fn(UpperCamelCase      ,			UpperCamelCase      ,			UpperCamelCase )
									lowerCamelCase__ :      str								=		rand_fn(UpperCamelCase      ,			UpperCamelCase      ,			UpperCamelCase )
									lowerCamelCase__ :      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``.
									lowerCamelCase__ :      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.
						lowerCamelCase__ :      List[Any]								=		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}''')








def      _A				(UpperCamelCase		:      List[Any]      ,			UpperCamelCase		:      Dict )  ->List[str]:



			'''simple docstring'''







			lowerCamelCase__ :      int								=		[1]
			for i in range(2      ,			UpperCamelCase ):
						factorials.append(factorials[-1] * i )
			assert 0 <= k < factorials[-1] * n, "k out of bounds"

			lowerCamelCase__ :      Union[str, Any]								=		[]
			lowerCamelCase__ :      Optional[Any]								=		list(range(UpperCamelCase ) )

			# Find permutation
			while factorials:
						lowerCamelCase__ :      Union[str, Any]								=		factorials.pop()
						lowerCamelCase__							,lowerCamelCase__ :      Union[str, Any]								=		divmod(UpperCamelCase      ,			UpperCamelCase )
						permutation.append(elements[number] )
						elements.remove(elements[number] )
			permutation.append(elements[0] )

			return permutation


if __name__ == "__main__":
				import doctest

				doctest.testmod()



