






from __future__ import annotations

from math import gcd






def 				lowerCamelCase__				(_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.
						SCREAMING_SNAKE_CASE						= seed
						SCREAMING_SNAKE_CASE						= seed

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

				a_		:						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',
				)
				a_		:						str             =    parser.parse_args()

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












from __future__ import annotations

import time

a_		:						Tuple             =    list[tuple[int, int]]

a_		:						Optional[int]             =    [
    [0, 0, 0, 0, 0, 0, 0],
    [0, 1, 0, 0, 0, 0, 0],  # 0 are free path whereas 1's are obstacles
    [0, 0, 0, 0, 0, 0, 0],
    [0, 0, 1, 0, 0, 0, 0],
    [1, 0, 1, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 1, 0, 0],
]

a_		:						Dict             =    [[-1, 0], [0, -1], [1, 0], [0, 1]]  # up, left, down, right

class 		_snake_case  :






			def __init__(       self		,					a		,					a		,					a		,					a		,					a)      ->       int:
						SCREAMING_SNAKE_CASE						= pos_x
						SCREAMING_SNAKE_CASE						= pos_y
						SCREAMING_SNAKE_CASE						= (pos_y, pos_x)
						SCREAMING_SNAKE_CASE						= goal_x
						SCREAMING_SNAKE_CASE						= goal_y
						SCREAMING_SNAKE_CASE						= parent



class 		_snake_case  :






			def __init__(       self		,					a		,					a)      ->       Dict:
						SCREAMING_SNAKE_CASE						= Node(start[1]		,					start[0]		,					goal[1]		,					goal[0]		,					a)
						SCREAMING_SNAKE_CASE						= Node(goal[1]		,					goal[0]		,					goal[1]		,					goal[0]		,					a)

						SCREAMING_SNAKE_CASE						= [self.start]
						SCREAMING_SNAKE_CASE						= False






			def    SCREAMING_SNAKE_CASE__    (       self)      ->       Path | None:
						while self.node_queue:
									SCREAMING_SNAKE_CASE						= self.node_queue.pop(0)

									if current_node.pos == self.target.pos:
												SCREAMING_SNAKE_CASE						= True
												return self.retrace_path(a)

									SCREAMING_SNAKE_CASE						= self.get_successors(a)

									for node in successors:
												self.node_queue.append(a)

						if not self.reached:
									return [self.start.pos]
						return None






			def    SCREAMING_SNAKE_CASE__    (       self		,					a)      ->       list[Node]:
						SCREAMING_SNAKE_CASE						= []
						for action in delta:
									SCREAMING_SNAKE_CASE						= parent.pos_x + action[1]
									SCREAMING_SNAKE_CASE						= parent.pos_y + action[0]
									if not (0 <= pos_x <= len(grid[0]) - 1 and 0 <= pos_y <= len(a) - 1):
												continue

									if grid[pos_y][pos_x] != 0:
												continue

									successors.append(
									    Node(a		,					a		,					self.target.pos_y		,					self.target.pos_x		,					a))
						return successors







			def    SCREAMING_SNAKE_CASE__    (       self		,					a)      ->       Path:
						SCREAMING_SNAKE_CASE						= node
						SCREAMING_SNAKE_CASE						= []
						while current_node is not None:
									path.append((current_node.pos_y, current_node.pos_x))
									SCREAMING_SNAKE_CASE						= current_node.parent
						path.reverse()
						return path



class 		_snake_case  :






			def __init__(       self		,					a		,					a)      ->       Tuple:
						SCREAMING_SNAKE_CASE						= BreadthFirstSearch(a		,					a)
						SCREAMING_SNAKE_CASE						= BreadthFirstSearch(a		,					a)
						SCREAMING_SNAKE_CASE						= False






			def    SCREAMING_SNAKE_CASE__    (       self)      ->       Path | None:
						while self.fwd_bfs.node_queue or self.bwd_bfs.node_queue:
									SCREAMING_SNAKE_CASE						= self.fwd_bfs.node_queue.pop(0)
									SCREAMING_SNAKE_CASE						= self.bwd_bfs.node_queue.pop(0)

									if current_bwd_node.pos == current_fwd_node.pos:
												SCREAMING_SNAKE_CASE						= True
												return self.retrace_bidirectional_path(
												    a		,					a)

									SCREAMING_SNAKE_CASE						= current_bwd_node
									SCREAMING_SNAKE_CASE						= current_fwd_node

									SCREAMING_SNAKE_CASE						= {
									    self.fwd_bfs: self.fwd_bfs.get_successors(a),
									    self.bwd_bfs: self.bwd_bfs.get_successors(a),
									}

									for bfs in [self.fwd_bfs, self.bwd_bfs]:
												for node in successors[bfs]:
															bfs.node_queue.append(a)

						if not self.reached:
									return [self.fwd_bfs.start.pos]
						return None







			def    SCREAMING_SNAKE_CASE__    (       self		,					a		,					a)      ->       Path:
						SCREAMING_SNAKE_CASE						= self.fwd_bfs.retrace_path(a)
						SCREAMING_SNAKE_CASE						= self.bwd_bfs.retrace_path(a)
						bwd_path.pop()
						bwd_path.reverse()
						SCREAMING_SNAKE_CASE						= fwd_path + bwd_path
						return path


if __name__ == "__main__":
				# all coordinates are given in format [y,x]
				import doctest

				doctest.testmod()
				a_		:						Union[str, Any]             =    (0, 0)
				a_		:						Any             =    (len(grid) - 1, len(grid[0]) - 1)
				for elem in grid:
								print(elem)

				a_		:						Optional[Any]             =    time.time()
				a_		:						Union[str, Any]             =    BreadthFirstSearch(init, goal)
				a_		:						Optional[Any]             =    bfs.search()
				a_		:						List[str]             =    time.time() - start_bfs_time

				print('Unidirectional BFS computation time : ', bfs_time)

				a_		:						Optional[int]             =    time.time()
				a_		:						Union[str, Any]             =    BidirectionalBreadthFirstSearch(init, goal)
				a_		:						List[Any]             =    bd_bfs.search()
				a_		:						List[str]             =    time.time() - start_bd_bfs_time

				print('Bidirectional BFS computation time : ', bd_bfs_time)




