


'''simple docstring'''

from __future__ import annotations

from math import gcd






def __UpperCamelCase   (			__lowerCamelCase		: Dict      ,  __lowerCamelCase		: int = 2      ,  __lowerCamelCase		: Optional[int] = 1      ,  __lowerCamelCase		: Union[str, Any] = 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(__lowerCamelCase		: List[str]      ,  __lowerCamelCase		: List[Any]      ,  __lowerCamelCase		: List[Any]				) -> int:
          return (pow(__lowerCamelCase      ,  2				) + step) % modulus

     for _ in range(__lowerCamelCase				):
          # 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(__lowerCamelCase      ,  __lowerCamelCase      ,  __lowerCamelCase				)
               _a						=		rand_fn(__lowerCamelCase      ,  __lowerCamelCase      ,  __lowerCamelCase				)
               _a						=		rand_fn(__lowerCamelCase      ,  __lowerCamelCase      ,  __lowerCamelCase				)

               # 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      ,  __lowerCamelCase				)

               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

   lowercase__				     =		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",
   )
   lowercase__				     =		parser.parse_args()

   lowercase__				     =		pollard_rho(args.num, attempts=args.attempts)
   if divisor is None:
      print(f'''{args.num} is probably prime''')
   else:
      lowercase__				     =		args.num // divisor
      print(f'''{args.num} = {divisor} * {quotient}''')





'''simple docstring'''

import warnings
from typing import List, Optional, Union

from ...processing_utils import ProcessorMixin
from ...tokenization_utils_base import BatchEncoding, PaddingStrategy, PreTokenizedInput, TextInput, TruncationStrategy
from ...utils import TensorType





class 						__SCREAMING_SNAKE_CASE					(      lowerCamelCase__ ):
     UpperCAmelCase 					=				['''image_processor''', '''tokenizer''']
     UpperCAmelCase 					=				'''LayoutLMv2ImageProcessor'''
     UpperCAmelCase 					=				('''LayoutXLMTokenizer''', '''LayoutXLMTokenizerFast''')






     def __init__(			self     ,							__UpperCamelCase=None     ,							__UpperCamelCase=None     ,							**__UpperCamelCase		)       ->    str:
          if "feature_extractor" in kwargs:
               warnings.warn(
                   "The `feature_extractor` argument is deprecated and will be removed in v5, use `image_processor`"
                   " instead."     ,							__UpperCamelCase     ,							)
               _a						=		kwargs.pop("feature_extractor"		)

          _a						=		image_processor if image_processor is not None else feature_extractor
          if image_processor is None:
               raise ValueError("You need to specify an `image_processor`."		)
          if tokenizer is None:
               raise ValueError("You need to specify a `tokenizer`."		)

          super().__init__(__UpperCamelCase     ,							__UpperCamelCase		)






     def __call__(			self     ,							__UpperCamelCase     ,							__UpperCamelCase = None     ,							__UpperCamelCase = None     ,							__UpperCamelCase = None     ,							__UpperCamelCase = None     ,							__UpperCamelCase = True     ,							__UpperCamelCase = False     ,							__UpperCamelCase = None     ,							__UpperCamelCase = None     ,							__UpperCamelCase = 0     ,							__UpperCamelCase = None     ,							__UpperCamelCase = None     ,							__UpperCamelCase = None     ,							__UpperCamelCase = False     ,							__UpperCamelCase = False     ,							__UpperCamelCase = False     ,							__UpperCamelCase = False     ,							__UpperCamelCase = True     ,							__UpperCamelCase = None     ,							**__UpperCamelCase     ,							)       ->    BatchEncoding:
          # verify input
          if self.image_processor.apply_ocr and (boxes is not None):
               raise ValueError(
                   "You cannot provide bounding boxes "
                   "if you initialized the image processor with apply_ocr set to True."		)

          if self.image_processor.apply_ocr and (word_labels is not None):
               raise ValueError(
                   "You cannot provide word labels if you initialized the image processor with apply_ocr set to True."		)

          if return_overflowing_tokens is True and return_offsets_mapping is False:
               raise ValueError("You cannot return overflowing tokens without returning the offsets mapping."		)

          # first, apply the image processor
          _a						=		self.image_processor(images=__UpperCamelCase     ,							return_tensors=__UpperCamelCase		)

          # second, apply the tokenizer
          if text is not None and self.image_processor.apply_ocr and text_pair is None:
               if isinstance(__UpperCamelCase     ,							__UpperCamelCase		):
                    _a						=		[text]  # add batch dimension (as the image processor always adds a batch dimension)
               _a						=		features["words"]

          _a						=		self.tokenizer(
              text=text if text is not None else features["words"]     ,							text_pair=text_pair if text_pair is not None else None     ,							boxes=boxes if boxes is not None else features["boxes"]     ,							word_labels=__UpperCamelCase     ,							add_special_tokens=__UpperCamelCase     ,							padding=__UpperCamelCase     ,							truncation=__UpperCamelCase     ,							max_length=__UpperCamelCase     ,							stride=__UpperCamelCase     ,							pad_to_multiple_of=__UpperCamelCase     ,							return_token_type_ids=__UpperCamelCase     ,							return_attention_mask=__UpperCamelCase     ,							return_overflowing_tokens=__UpperCamelCase     ,							return_special_tokens_mask=__UpperCamelCase     ,							return_offsets_mapping=__UpperCamelCase     ,							return_length=__UpperCamelCase     ,							verbose=__UpperCamelCase     ,							return_tensors=__UpperCamelCase     ,							**__UpperCamelCase     ,							)

          # add pixel values
          _a						=		features.pop("pixel_values"		)
          if return_overflowing_tokens is True:
               _a						=		self.get_overflowing_images(__UpperCamelCase     ,							encoded_inputs["overflow_to_sample_mapping"]		)
          _a						=		images

          return encoded_inputs






     def  a_					(			self     ,							__UpperCamelCase     ,							__UpperCamelCase		)       ->    Dict:
          # in case there's an overflow, ensure each `input_ids` sample is mapped to its corresponding image
          _a						=		[]
          for sample_idx in overflow_to_sample_mapping:
               images_with_overflow.append(images[sample_idx]		)

          if len(__UpperCamelCase		) != len(__UpperCamelCase		):
               raise ValueError(
                   "Expected length of images to be the same as the length of `overflow_to_sample_mapping`, but got"
                   f" {len(__UpperCamelCase		)} and {len(__UpperCamelCase		)}"		)

          return images_with_overflow






     def  a_					(			self     ,							*__UpperCamelCase     ,							**__UpperCamelCase		)       ->    Tuple:
          return self.tokenizer.batch_decode(*__UpperCamelCase     ,							**__UpperCamelCase		)






     def  a_					(			self     ,							*__UpperCamelCase     ,							**__UpperCamelCase		)       ->    List[Any]:
          return self.tokenizer.decode(*__UpperCamelCase     ,							**__UpperCamelCase		)






     @property
     def  a_					(			self		)       ->    int:
          return ["input_ids", "bbox", "attention_mask", "image"]






     @property
     def  a_					(			self		)       ->    Dict:
          warnings.warn(
              "`feature_extractor_class` is deprecated and will be removed in v5. Use `image_processor_class` instead."     ,							__UpperCamelCase     ,							)
          return self.image_processor_class






     @property
     def  a_					(			self		)       ->    int:
          warnings.warn(
              "`feature_extractor` is deprecated and will be removed in v5. Use `image_processor` instead."     ,							__UpperCamelCase     ,							)
          return self.image_processor

