




from __future__ import annotations

from math import gcd




def  SCREAMING_SNAKE_CASE     (	__UpperCamelCase      :						int	,					__UpperCamelCase      :						int = 2	,					__UpperCamelCase      :						int = 1	,					__UpperCamelCase      :						int = 3	,					)  ->							int | None:
    # 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      :						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_										=					seed
        UpperCAmelCase_										=					seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            UpperCAmelCase_										=					rand_fn(__UpperCamelCase	,					__UpperCamelCase	,					__UpperCamelCase				)
            UpperCAmelCase_										=					rand_fn(__UpperCamelCase	,					__UpperCamelCase	,					__UpperCamelCase				)
            UpperCAmelCase_										=					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_										=					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_										=					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 typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_flax_available,
    is_tf_available,
    is_tokenizers_available,
    is_torch_available,
)


_lowerCamelCase      			=						{
    'configuration_roberta': ['ROBERTA_PRETRAINED_CONFIG_ARCHIVE_MAP', 'RobertaConfig', 'RobertaOnnxConfig'],
    'tokenization_roberta': ['RobertaTokenizer'],
}

try:
 if not is_tokenizers_available():
  raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
 pass
else:
 _lowerCamelCase      			=						['RobertaTokenizerFast']

try:
 if not is_torch_available():
  raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
 pass
else:
 _lowerCamelCase      			=						[
     'ROBERTA_PRETRAINED_MODEL_ARCHIVE_LIST',
     'RobertaForCausalLM',
     'RobertaForMaskedLM',
     'RobertaForMultipleChoice',
     'RobertaForQuestionAnswering',
     'RobertaForSequenceClassification',
     'RobertaForTokenClassification',
     'RobertaModel',
     'RobertaPreTrainedModel',
 ]

try:
 if not is_tf_available():
  raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
 pass
else:
 _lowerCamelCase      			=						[
     'TF_ROBERTA_PRETRAINED_MODEL_ARCHIVE_LIST',
     'TFRobertaForCausalLM',
     'TFRobertaForMaskedLM',
     'TFRobertaForMultipleChoice',
     'TFRobertaForQuestionAnswering',
     'TFRobertaForSequenceClassification',
     'TFRobertaForTokenClassification',
     'TFRobertaMainLayer',
     'TFRobertaModel',
     'TFRobertaPreTrainedModel',
 ]

try:
 if not is_flax_available():
  raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
 pass
else:
 _lowerCamelCase      			=						[
     'FlaxRobertaForCausalLM',
     'FlaxRobertaForMaskedLM',
     'FlaxRobertaForMultipleChoice',
     'FlaxRobertaForQuestionAnswering',
     'FlaxRobertaForSequenceClassification',
     'FlaxRobertaForTokenClassification',
     'FlaxRobertaModel',
     'FlaxRobertaPreTrainedModel',
 ]


if TYPE_CHECKING:
 from .configuration_roberta import ROBERTA_PRETRAINED_CONFIG_ARCHIVE_MAP, RobertaConfig, RobertaOnnxConfig
 from .tokenization_roberta import RobertaTokenizer

 try:
  if not is_tokenizers_available():
   raise OptionalDependencyNotAvailable()
 except OptionalDependencyNotAvailable:
  pass
 else:
  from .tokenization_roberta_fast import RobertaTokenizerFast

 try:
  if not is_torch_available():
   raise OptionalDependencyNotAvailable()
 except OptionalDependencyNotAvailable:
  pass
 else:
  from .modeling_roberta import (
      ROBERTA_PRETRAINED_MODEL_ARCHIVE_LIST,
      RobertaForCausalLM,
      RobertaForMaskedLM,
      RobertaForMultipleChoice,
      RobertaForQuestionAnswering,
      RobertaForSequenceClassification,
      RobertaForTokenClassification,
      RobertaModel,
      RobertaPreTrainedModel,
  )

 try:
  if not is_tf_available():
   raise OptionalDependencyNotAvailable()
 except OptionalDependencyNotAvailable:
  pass
 else:
  from .modeling_tf_roberta import (
      TF_ROBERTA_PRETRAINED_MODEL_ARCHIVE_LIST,
      TFRobertaForCausalLM,
      TFRobertaForMaskedLM,
      TFRobertaForMultipleChoice,
      TFRobertaForQuestionAnswering,
      TFRobertaForSequenceClassification,
      TFRobertaForTokenClassification,
      TFRobertaMainLayer,
      TFRobertaModel,
      TFRobertaPreTrainedModel,
  )

 try:
  if not is_flax_available():
   raise OptionalDependencyNotAvailable()
 except OptionalDependencyNotAvailable:
  pass
 else:
  from .modeling_flax_roberta import (
      FlaxRobertaForCausalLM,
      FlaxRobertaForMaskedLM,
      FlaxRobertaForMultipleChoice,
      FlaxRobertaForQuestionAnswering,
      FlaxRobertaForSequenceClassification,
      FlaxRobertaForTokenClassification,
      FlaxRobertaModel,
      FlaxRobertaPreTrainedModel,
  )

else:
 import sys

 _lowerCamelCase      			=						_LazyModule(__name__, globals()['__file__'], _import_structure, module_spec=__spec__)





