





from __future__ import annotations

from math import gcd

def   a_    ( lowerCAmelCase_      :			int,	lowerCAmelCase_      :			int = 2,	lowerCAmelCase_      :			int = 1,	lowerCAmelCase_      :			int = 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(lowerCAmelCase_      :			int,	lowerCAmelCase_      :			int,	lowerCAmelCase_      :			int	) -> int:
      return (pow(__UpperCAmelCase,	2	) + step) % modulus

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

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

     _snake_case  :    List[str]						    =  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',
     )
     _snake_case  :    Tuple						    =  parser.parse_args()

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







from typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_tokenizers_available,
    is_torch_available,
    is_vision_available,
)


lowerCamelCase__ :   Union[str, Any]					=							{
    'configuration_layoutlmv2': ['LAYOUTLMV2_PRETRAINED_CONFIG_ARCHIVE_MAP', 'LayoutLMv2Config'],
    'processing_layoutlmv2': ['LayoutLMv2Processor'],
    'tokenization_layoutlmv2': ['LayoutLMv2Tokenizer'],
}

try:
   if not is_tokenizers_available():
      raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
   pass
else:
   lowerCamelCase__ :   Any					=							['LayoutLMv2TokenizerFast']

try:
   if not is_vision_available():
      raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
   pass
else:
   lowerCamelCase__ :   Dict					=							['LayoutLMv2FeatureExtractor']
   lowerCamelCase__ :   Dict					=							['LayoutLMv2ImageProcessor']

try:
   if not is_torch_available():
      raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
   pass
else:
   lowerCamelCase__ :   Union[str, Any]					=							[
       'LAYOUTLMV2_PRETRAINED_MODEL_ARCHIVE_LIST',
       'LayoutLMv2ForQuestionAnswering',
       'LayoutLMv2ForSequenceClassification',
       'LayoutLMv2ForTokenClassification',
       'LayoutLMv2Layer',
       'LayoutLMv2Model',
       'LayoutLMv2PreTrainedModel',
   ]

if TYPE_CHECKING:
   from .configuration_layoutlmva import LAYOUTLMV2_PRETRAINED_CONFIG_ARCHIVE_MAP, LayoutLMvaConfig
   from .processing_layoutlmva import LayoutLMvaProcessor
   from .tokenization_layoutlmva import LayoutLMvaTokenizer

   try:
      if not is_tokenizers_available():
         raise OptionalDependencyNotAvailable()
   except OptionalDependencyNotAvailable:
      pass
   else:
      from .tokenization_layoutlmva_fast import LayoutLMvaTokenizerFast

   try:
      if not is_vision_available():
         raise OptionalDependencyNotAvailable()
   except OptionalDependencyNotAvailable:
      pass
   else:
      from .feature_extraction_layoutlmva import LayoutLMvaFeatureExtractor, LayoutLMvaImageProcessor

   try:
      if not is_torch_available():
         raise OptionalDependencyNotAvailable()
   except OptionalDependencyNotAvailable:
      pass
   else:
      from .modeling_layoutlmva import (
          LAYOUTLMV2_PRETRAINED_MODEL_ARCHIVE_LIST,
          LayoutLMvaForQuestionAnswering,
          LayoutLMvaForSequenceClassification,
          LayoutLMvaForTokenClassification,
          LayoutLMvaLayer,
          LayoutLMvaModel,
          LayoutLMvaPreTrainedModel,
      )
else:
   import sys

   lowerCamelCase__ :   Any					=							_LazyModule(__name__, globals()['__file__'], _import_structure, module_spec=__spec__)