


"""simple docstring"""



from __future__ import annotations

from collections import deque
from enum import Enum
from math import atana, degrees
from sys import maxsize







class       __a	(   __snake_case       ):
				lowerCamelCase    :					List[str] 					=1
				lowerCamelCase    :					int 					=2
				lowerCamelCase    :					List[Any] 					=3







				def __repr__(   self      ):

					'''simple docstring'''






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




def     UpperCAmelCase    (					_lowercase   :      tuple[int, int] ,      _lowercase   :      int ,      _lowercase   :      int  )	->   float:


	"""simple docstring"""




	lowerCAmelCase_       ,  lowerCAmelCase_			       = point
	return degrees(atana(y - miny ,      x - minx  )  )
def     UpperCAmelCase    (					_lowercase   :      tuple[int, int] ,      _lowercase   :      tuple[int, int] ,      _lowercase   :      tuple[int, int]  )	->   Direction:


	"""simple docstring"""




	lowerCAmelCase_       ,  lowerCAmelCase_			       = starting
	lowerCAmelCase_       ,  lowerCAmelCase_			       = via
	lowerCAmelCase_       ,  lowerCAmelCase_			       = target
	lowerCAmelCase_			       = degrees(atana(ya - ya ,      xa - xa  )  )
	via_angle %= 3_6_0
	lowerCAmelCase_			       = degrees(atana(ya - ya ,      xa - xa  )  )
	target_angle %= 3_6_0
	# 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     UpperCAmelCase    (					_lowercase   :      list[tuple[int, int]]  )	->   list[tuple[int, int]]:


	"""simple docstring"""




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

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

	lowerCAmelCase_			       = sorted(_lowercase ,      key=lambda _lowercase  : angle_comparer(_lowercase ,      _lowercase ,      _lowercase  )  )
	# 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)  )

	lowerCAmelCase_			       = 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.
	lowerCAmelCase_			       = Direction.left

	for i in range(3 ,      len(_lowercase  )  ):
		while True:
			lowerCAmelCase_			       = stack[-2]
			lowerCAmelCase_			       = stack[-1]
			lowerCAmelCase_			       = sorted_points[i]
			lowerCAmelCase_			       = check_direction(_lowercase ,      _lowercase ,      _lowercase  )

			if next_direction == Direction.left:
				lowerCAmelCase_			       = 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(_lowercase  )



"""simple docstring"""



def     UpperCAmelCase    (					_lowercase   :      list[int]  )	->   int:


	"""simple docstring"""




	lowerCAmelCase_			       = len(_lowercase  )
	if n in (0, 1):
		return 0
	return sum(query * (n - i - 1) for i, query in enumerate(sorted(_lowercase  )  )  )


if __name__ == "__main__":
		import doctest

		doctest.testmod()