

from __future__ import annotations

from math import gcd


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

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

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

																		# 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``.
																		snake_case_   =	gcd(hare - tortoise  ,		SCREAMING_SNAKE_CASE__       )

																		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.
												snake_case_   =	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 decimal import Decimal, getcontext
from math import ceil, factorial


def   __SCREAMING_SNAKE_CASE					(SCREAMING_SNAKE_CASE__       ):

						if not isinstance(SCREAMING_SNAKE_CASE__  ,		SCREAMING_SNAKE_CASE__       ):
												raise TypeError('''Undefined for non-integers'''       )
						elif precision < 1:
												raise ValueError('''Undefined for non-natural numbers'''       )

						snake_case_   =	precision
						snake_case_   =	ceil(precision / 14       )
						snake_case_   =	426880 * Decimal(10005       ).sqrt()
						snake_case_   =	1
						snake_case_   =	13591409
						snake_case_   =	Decimal(SCREAMING_SNAKE_CASE__       )
						for k in range(1  ,		SCREAMING_SNAKE_CASE__       ):
												snake_case_   =	factorial(6 * k       ) // (factorial(3 * k       ) * factorial(SCREAMING_SNAKE_CASE__       ) ** 3)
												linear_term += 545140134
												exponential_term *= -262537412640768000
												partial_sum += Decimal(multinomial_term * linear_term       ) / exponential_term
						return str(constant_term / partial_sum       )[:-1]


if __name__ == "__main__":
					lowerCAmelCase_       	=	50
					print(f"""The first {n} digits of pi is: {pi(n)}""")