



"""simple docstring"""






from __future__ import annotations

from math import gcd






def 	snake_case_  (				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.
              _lowerCamelCase						:    Dict         =			seed
              _lowerCamelCase						:    Dict         =			seed

              while True:
                     # At each iteration, the tortoise moves one step and the hare moves two.
                     _lowerCamelCase						:    Union[str, Any]         =			rand_fn(A_,    A_,    A_						)
                     _lowerCamelCase						:    Optional[int]         =			rand_fn(A_,    A_,    A_						)
                     _lowerCamelCase						:    str         =			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``.
                     _lowerCamelCase						:    Optional[Any]         =			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.
              _lowerCamelCase						:    List[str]         =			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}""")







"""simple docstring"""






def 	snake_case_  (				A_ :		list						):


       '''simple docstring'''





       _lowerCamelCase						:    Union[str, Any]         =			len(A_						)
       for i in range(1,    A_						):
              _lowerCamelCase						:    Tuple         =			collection[i]
              _lowerCamelCase						:    Dict         =			0
              _lowerCamelCase						:    Any         =			i - 1

              while low <= high:
                     _lowerCamelCase						:    Optional[int]         =			(low + high) // 2
                     if val < collection[mid]:
                            _lowerCamelCase						:    List[str]         =			mid - 1
                     else:
                            _lowerCamelCase						:    Dict         =			mid + 1
              for j in range(A_,    A_,    -1						):
                     _lowerCamelCase						:    Optional[int]         =			collection[j - 1]
              _lowerCamelCase						:    Tuple         =			val
       return collection


if __name__ == "__main__":
   lowerCAmelCase__      =    input('''Enter numbers separated by a comma:\n''').strip()
   lowerCAmelCase__      =    [int(item) for item in user_input.split(''',''')]
   print(binary_insertion_sort(unsorted))


