from __future__ import annotations

from math import gcd


def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase = 2     ,__UpperCAmelCase = 1     ,__UpperCAmelCase = 3     ,):






	"""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     ,__UpperCAmelCase     ,__UpperCAmelCase			) -> int:
		return (pow(__UpperCAmelCase     ,2			) + step) % modulus

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

		while True:
			# At each iteration, the tortoise moves one step and the hare moves two.
			_lowercase :    Tuple							=							rand_fn(__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			)
			_lowercase :    str							=							rand_fn(__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			)
			_lowercase :    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``.
			_lowercase :    str							=							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.
		_lowercase :    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

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

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




"""simple docstring"""







def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	_lowercase :    Any							=							len(__UpperCAmelCase			)
	_lowercase :    Any							=							len(__UpperCAmelCase			)
	_lowercase :    List[Any]							=							[[False for _ in range(m + 1			)] for _ in range(n + 1			)]
	_lowercase :    Union[str, Any]							=							True
	for i in range(__UpperCAmelCase			):
		for j in range(m + 1			):
			if dp[i][j]:
				if j < m and a[i].upper() == b[j]:
					_lowercase :    Optional[int]							=							True
				if a[i].islower():
					_lowercase :    Union[str, Any]							=							True
	return dp[n][m]


if __name__ == "__main__":
						import doctest

						doctest.testmod()



