'''simple docstring'''

from __future__ import annotations

from math import gcd
def 			__UpperCamelCase       (   UpperCAmelCase		,     UpperCAmelCase = 2		,     UpperCAmelCase = 1		,     UpperCAmelCase = 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(UpperCAmelCase		,     UpperCAmelCase		,     UpperCAmelCase     ) -> int:
  return (pow(__UpperCamelCase		,     2     ) + step) % modulus

 for _ in range(__UpperCamelCase     ):
  # These track the position within the cycle detection logic.
  lowercase__				:    Any  =					seed
  lowercase__				:    Dict  =					seed

  while True:
   # At each iteration, the tortoise moves one step and the hare moves two.
   lowercase__				:    List[str]  =					rand_fn(__UpperCamelCase		,     __UpperCamelCase		,     __UpperCamelCase     )
   lowercase__				:    List[str]  =					rand_fn(__UpperCamelCase		,     __UpperCamelCase		,     __UpperCamelCase     )
   lowercase__				:    Tuple  =					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``.
   lowercase__				:    Optional[Any]  =					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.
  lowercase__				:    Tuple  =					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

  __a: Dict          =   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""",
  )
  __a: Dict          =   parser.parse_args()

  __a: Union[str, Any]          =   pollard_rho(args.num, attempts=args.attempts)
  if divisor is None:
    print(F'{args.num} is probably prime')
  else:
    __a: Optional[Any]          =   args.num // divisor
    print(F'{args.num} = {divisor} * {quotient}')


import math
from datetime import datetime, timedelta


def    a__       (						__UpperCamelCase ):
      SCREAMING_SNAKE_CASE_      =     year % 1_9
      SCREAMING_SNAKE_CASE_      =     year % 4
      SCREAMING_SNAKE_CASE_      =     year % 7
      SCREAMING_SNAKE_CASE_      =     math.floor(year / 1_0_0 )
      SCREAMING_SNAKE_CASE_      =     math.floor((1_3 + 8 * leap_day_inhibits) / 2_5 )
      SCREAMING_SNAKE_CASE_      =     leap_day_inhibits / 4
      SCREAMING_SNAKE_CASE_      =     (
          1_5 - lunar_orbit_correction + leap_day_inhibits - leap_day_reinstall_number
      ) % 3_0
      SCREAMING_SNAKE_CASE_      =     (4 + leap_day_inhibits - leap_day_reinstall_number) % 7

      # days to be added to March 21
      SCREAMING_SNAKE_CASE_      =     (1_9 * metonic_cycle + secular_moon_shift) % 3_0

      # PHM -> Paschal Full Moon
      SCREAMING_SNAKE_CASE_      =     (
          2 * julian_leap_year
          + 4 * non_leap_year
          + 6 * days_to_add
          + century_starting_point
      ) % 7

      if days_to_add == 2_9 and days_from_phm_to_sunday == 6:
            return datetime(__UpperCamelCase   ,					4   ,					1_9 )
      elif days_to_add == 2_8 and days_from_phm_to_sunday == 6:
            return datetime(__UpperCamelCase   ,					4   ,					1_8 )
      else:
            return datetime(__UpperCamelCase   ,					3   ,					2_2 ) + timedelta(
                days=int(days_to_add + days_from_phm_to_sunday ) )


if __name__ == "__main__":
      for year in (19_94, 20_00, 20_10, 20_21, 20_23):
            A   :     Dict       =				"will be" if year > datetime.now().year else "was"
            print(f"Easter in {year} {tense} {gauss_easter(year)}")


