




"""simple docstring"""




from __future__ import annotations

from math import gcd







def      __UpperCAmelCase     ( __UpperCamelCase	,				__UpperCamelCase = 2	,				__UpperCamelCase = 1	,				__UpperCamelCase = 3	,				):
			# A value less than 2 can cause an infinite loop in the algorithm.
			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			:  List[str]								=       seed
						__lowercase			:  Tuple								=       seed

						while True:
									# At each iteration, the tortoise moves one step and the hare moves two.
									__lowercase			:  Optional[Any]								=       rand_fn(__UpperCamelCase	,				__UpperCamelCase	,				__UpperCamelCase	)
									__lowercase			:  Optional[int]								=       rand_fn(__UpperCamelCase	,				__UpperCamelCase	,				__UpperCamelCase	)
									__lowercase			:  Any								=       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			:  List[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.
						__lowercase			:  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

			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'''

import argparse
from collections import defaultdict

import yaml


a    :							List[Any]										=    'docs/source/en/_toctree.yml'


def 						__magic_name__       (      __UpperCAmelCase      )						->   str:

					'''simple docstring'''

					snake_case_		     =							defaultdict(__UpperCAmelCase      )
					for doc in model_doc:
										counts[doc["local"]] += 1
					snake_case_		     =							[key for key, value in counts.items() if value > 1]

					snake_case_		     =							[]
					for duplicate_key in duplicates:
										snake_case_		     =							list({doc['''title'''] for doc in model_doc if doc['''local'''] == duplicate_key}      )
										if len(__UpperCAmelCase      ) > 1:
															raise ValueError(
															    F"{duplicate_key} is present several times in the documentation table of content at "
															    '''`docs/source/en/_toctree.yml` with different *Title* values. Choose one of those and remove the '''
															    '''others.'''      )
										# Only add this once
										new_doc.append({'''local''': duplicate_key, '''title''': titles[0]}      )

					# Add none duplicate-keys
					new_doc.extend([doc for doc in model_doc if counts[doc['''local''']] == 1]      )

					# Sort
					return sorted(__UpperCAmelCase, key=lambda __UpperCAmelCase      : s["title"].lower()      )


def 						__magic_name__       (      __UpperCAmelCase=False      )						->   List[Any]:

					'''simple docstring'''

					with open(__UpperCAmelCase, encoding='''utf-8'''      ) as f:
										snake_case_		     =							yaml.safe_load(f.read()      )

					# Get to the API doc
					snake_case_		     =							0
					while content[api_idx]["title"] != "API":
										api_idx += 1
					snake_case_		     =							content[api_idx]['''sections''']

					# Then to the model doc
					snake_case_		     =							0
					while api_doc[model_idx]["title"] != "Models":
										model_idx += 1

					snake_case_		     =							api_doc[model_idx]['''sections''']

					snake_case_		     =							[(idx, section) for idx, section in enumerate(__UpperCAmelCase      ) if '''sections''' in section]
					snake_case_		     =							False
					for idx, modality_doc in modalities_docs:
										snake_case_		     =							modality_doc['''sections''']
										snake_case_		     =							clean_model_doc_toc(__UpperCAmelCase      )

										if old_modality_doc != new_modality_doc:
															snake_case_		     =							True
															if overwrite:
																				snake_case_		     =							new_modality_doc

					if diff:
										if overwrite:
															snake_case_		     =							model_doc
															snake_case_		     =							api_doc
															with open(__UpperCAmelCase, '''w''', encoding='''utf-8'''      ) as f:
																				f.write(yaml.dump(__UpperCAmelCase, allow_unicode=__UpperCAmelCase      )      )
										else:
															raise ValueError(
															    '''The model doc part of the table of content is not properly sorted, run `make style` to fix this.'''      )


if __name__ == "__main__":
							a    :							Dict										=    argparse.ArgumentParser()
							parser.add_argument('--fix_and_overwrite', action='store_true', help='Whether to fix inconsistencies.')
							a    :							Union[str, Any]										=    parser.parse_args()

							check_model_doc(args.fix_and_overwrite)

