
"""simple docstring"""





from __future__ import annotations

from math import gcd



def 				lowerCamelCase__		(      __snake_case,					__snake_case = 2,					__snake_case = 1,					__snake_case = 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(__snake_case,					__snake_case,					__snake_case     ) -> int:
										return (pow(__snake_case,					2     ) + step) % modulus

					for _ in range(__snake_case     ):
										# These track the position within the cycle detection logic.
										_UpperCamelCase										=  seed
										_UpperCamelCase										=  seed

										while True:
															# At each iteration, the tortoise moves one step and the hare moves two.
															_UpperCamelCase										=  rand_fn(__snake_case,					__snake_case,					__snake_case     )
															_UpperCamelCase										=  rand_fn(__snake_case,					__snake_case,					__snake_case     )
															_UpperCamelCase										=  rand_fn(__snake_case,					__snake_case,					__snake_case     )

															# 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										=  gcd(hare - tortoise,					__snake_case     )

															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										=  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



def 				lowerCamelCase__		(      __snake_case,					__snake_case     )							->		Optional[int]:


					"""simple docstring"""





					if len(__snake_case     ) <= 1 or n <= 1:
										return

					insert_next(__snake_case,					n - 1     )
					rec_insertion_sort(__snake_case,					n - 1     )



def 				lowerCamelCase__		(      __snake_case,					__snake_case     )							->		Dict:


					"""simple docstring"""





					if index >= len(__snake_case     ) or collection[index - 1] <= collection[index]:
										return

					# Swaps adjacent elements since they are not in ascending order
					_UpperCamelCase			,  _UpperCamelCase										=  (
					    collection[index],
					    collection[index - 1],
					)

					insert_next(__snake_case,					index + 1     )


if __name__ == "__main__":
				_a														=		input("""Enter integers separated by spaces: """)
				_a														=		[int(num) for num in numbers.split()]
				rec_insertion_sort(number_list, len(number_list))
				print(number_list)






