



"""simple docstring"""

from __future__ import annotations

from collections import deque
from enum import Enum
from math import atana, degrees
from sys import maxsize
class 						lowerCAmelCase__	(  _lowerCamelCase   ):


	'''simple docstring'''







	_SCREAMING_SNAKE_CASE  :		Optional[Any]     	=   1
	_SCREAMING_SNAKE_CASE  :		Tuple     	=   2
	_SCREAMING_SNAKE_CASE  :		Dict     	=   3





	def __repr__(			self							:					int						)	->  int:





					"""simple docstring"""


					return f"""{self.__class__.__name__}.{self.name}"""




def    __snake_case       (  __A			:	tuple[int, int]       ,		__A			:	int       ,		__A			:	int    )    ->					float:






				'''simple docstring'''
				SCREAMING_SNAKE_CASE	:       Union[str, Any] 							=    point
				return degrees(atana(y - miny       ,		x - minx    )    )
def    __snake_case       (  __A			:	tuple[int, int]       ,		__A			:	tuple[int, int]       ,		__A			:	tuple[int, int]    )    ->					Direction:






				'''simple docstring'''
				SCREAMING_SNAKE_CASE	:       Any 							=    starting
				SCREAMING_SNAKE_CASE	:       str 							=    via
				SCREAMING_SNAKE_CASE	:       List[Any] 							=    target
				SCREAMING_SNAKE_CASE	:       str 							=    degrees(atana(ya - ya       ,		xa - xa    )    )
				via_angle %= 360
				SCREAMING_SNAKE_CASE	:       Any 							=    degrees(atana(ya - ya       ,		xa - xa    )    )
				target_angle %= 360
				# t-
				#  \ \
				#   \ v
				#    \|
				#     s
				# via_angle is always lower than target_angle, if direction is left.
				# If they are same, it means they are on a same line of convex hull.
				if target_angle > via_angle:
								return Direction.left
				elif target_angle == via_angle:
								return Direction.straight
				else:
								return Direction.right







def    __snake_case       (  __A			:	list[tuple[int, int]]    )    ->					list[tuple[int, int]]:






				'''simple docstring'''

				if len(__A    ) <= 2:
								# There is no convex hull
								raise ValueError('graham_scan: argument must contain more than 3 points.'    )
				if len(__A    ) == 3:
								return points
				# find the lowest and the most left point
				SCREAMING_SNAKE_CASE	:       Dict 							=    0
				SCREAMING_SNAKE_CASE	:       Any 							=    maxsize, maxsize
				for i, point in enumerate(__A    ):
								SCREAMING_SNAKE_CASE	:       Optional[int] 							=    point[0]
								SCREAMING_SNAKE_CASE	:       str 							=    point[1]
								if y < miny:
												SCREAMING_SNAKE_CASE	:       List[Any] 							=    y
												SCREAMING_SNAKE_CASE	:       Tuple 							=    x
												SCREAMING_SNAKE_CASE	:       Any 							=    i
								if y == miny and x < minx:
												SCREAMING_SNAKE_CASE	:       Dict 							=    x
												SCREAMING_SNAKE_CASE	:       Union[str, Any] 							=    i

    # remove the lowest and the most left point from points for preparing for sort
				points.pop(__A    )

				SCREAMING_SNAKE_CASE	:       Tuple 							=    sorted(__A       ,		key=lambda __A    : angle_comparer(__A       ,		__A       ,		__A    )    )
				# This insert actually costs complexity,
				# and you should instead add (minx, miny) into stack later.
				# I'm using insert just for easy understanding.
				sorted_points.insert(0       ,		(minx, miny)    )

				SCREAMING_SNAKE_CASE	:       deque[tuple[int, int]] 							=    deque()
				stack.append(sorted_points[0]    )
				stack.append(sorted_points[1]    )
				stack.append(sorted_points[2]    )
				# In any ways, the first 3 points line are towards left.
				# Because we sort them the angle from minx, miny.
				SCREAMING_SNAKE_CASE	:       List[str] 							=    Direction.left

				for i in range(3       ,		len(__A    )    ):
								while True:
												SCREAMING_SNAKE_CASE	:       Dict 							=    stack[-2]
												SCREAMING_SNAKE_CASE	:       Any 							=    stack[-1]
												SCREAMING_SNAKE_CASE	:       List[Any] 							=    sorted_points[i]
												SCREAMING_SNAKE_CASE	:       Tuple 							=    check_direction(__A       ,		__A       ,		__A    )

												if next_direction == Direction.left:
																SCREAMING_SNAKE_CASE	:       Union[str, Any] 							=    Direction.left
																break
												if next_direction == Direction.straight:
																if current_direction == Direction.left:
																				# We keep current_direction as left.
																				# Because if the straight line keeps as straight,
																				# we want to know if this straight line is towards left.
																				break
																elif current_direction == Direction.right:
																				# If the straight line is towards right,
																				# every previous points on those straigh line is not convex hull.
																				stack.pop()
												if next_direction == Direction.right:
																stack.pop()
								stack.append(sorted_points[i]    )
				return list(__A    )






"""simple docstring"""

def    __snake_case       (  __A			:	int    )    ->					int:






				'''simple docstring'''
				if not isinstance(__A       ,		__A    ):
								raise ValueError('double_factorial() only accepts integral values'    )
				if num < 0:
								raise ValueError('double_factorial() not defined for negative values'    )
				SCREAMING_SNAKE_CASE	:       List[str] 							=    1
				for i in range(__A       ,		0       ,		-2    ):
								value *= i
				return value


if __name__ == "__main__":
							import doctest

							doctest.testmod()





