

from __future__ import annotations

from math import gcd




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

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

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

      # 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``.
      a__  =gcd(hare - tortoise	,      _A							)

      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.
    a__  =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_tf_available,
    is_tokenizers_available,
    is_torch_available,
)


lowerCamelCase           =					{
    '''configuration_deberta''': ['''DEBERTA_PRETRAINED_CONFIG_ARCHIVE_MAP''', '''DebertaConfig''', '''DebertaOnnxConfig'''],
    '''tokenization_deberta''': ['''DebertaTokenizer'''],
}

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

try:
       if not is_torch_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       lowerCamelCase           =					[
           '''DEBERTA_PRETRAINED_MODEL_ARCHIVE_LIST''',
           '''DebertaForMaskedLM''',
           '''DebertaForQuestionAnswering''',
           '''DebertaForSequenceClassification''',
           '''DebertaForTokenClassification''',
           '''DebertaModel''',
           '''DebertaPreTrainedModel''',
       ]

try:
       if not is_tf_available():
              raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
       pass
else:
       lowerCamelCase           =					[
           '''TF_DEBERTA_PRETRAINED_MODEL_ARCHIVE_LIST''',
           '''TFDebertaForMaskedLM''',
           '''TFDebertaForQuestionAnswering''',
           '''TFDebertaForSequenceClassification''',
           '''TFDebertaForTokenClassification''',
           '''TFDebertaModel''',
           '''TFDebertaPreTrainedModel''',
       ]


if TYPE_CHECKING:
       from .configuration_deberta import DEBERTA_PRETRAINED_CONFIG_ARCHIVE_MAP, DebertaConfig, DebertaOnnxConfig
       from .tokenization_deberta import DebertaTokenizer

       try:
              if not is_tokenizers_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .tokenization_deberta_fast import DebertaTokenizerFast

       try:
              if not is_torch_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .modeling_deberta import (
                  DEBERTA_PRETRAINED_MODEL_ARCHIVE_LIST,
                  DebertaForMaskedLM,
                  DebertaForQuestionAnswering,
                  DebertaForSequenceClassification,
                  DebertaForTokenClassification,
                  DebertaModel,
                  DebertaPreTrainedModel,
              )

       try:
              if not is_tf_available():
                     raise OptionalDependencyNotAvailable()
       except OptionalDependencyNotAvailable:
              pass
       else:
              from .modeling_tf_deberta import (
                  TF_DEBERTA_PRETRAINED_MODEL_ARCHIVE_LIST,
                  TFDebertaForMaskedLM,
                  TFDebertaForQuestionAnswering,
                  TFDebertaForSequenceClassification,
                  TFDebertaForTokenClassification,
                  TFDebertaModel,
                  TFDebertaPreTrainedModel,
              )


else:
       import sys

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