




from __future__ import annotations

from collections.abc import Iterable




class 	UpperCamelCase__			:

				'''simple docstring'''



				def __init__(					self  ,						UpperCamelCase__  ,						UpperCamelCase__  )  ->   str:
											lowerCamelCase     :   Union[str, Any]    				=      float(UpperCamelCase__  ), float(UpperCamelCase__  )







				def __eq__(					self  ,						UpperCamelCase__  )  ->   List[Any]:
											return self.x == other.x and self.y == other.y







				def __ne__(					self  ,						UpperCamelCase__  )  ->   Tuple:
											return not self == other







				def __gt__(					self  ,						UpperCamelCase__  )  ->   Optional[Any]:
											if self.x > other.x:
																		return True
											elif self.x == other.x:
																		return self.y > other.y
											return False







				def __lt__(					self  ,						UpperCamelCase__  )  ->   List[str]:
											return not self > other







				def __ge__(					self  ,						UpperCamelCase__  )  ->   Union[str, Any]:
											if self.x > other.x:
																		return True
											elif self.x == other.x:
																		return self.y >= other.y
											return False







				def __le__(					self  ,						UpperCamelCase__  )  ->   int:
											if self.x < other.x:
																		return True
											elif self.x == other.x:
																		return self.y <= other.y
											return False







				def __repr__(					self  )  ->   str:
											return F'''({self.x}, {self.y})'''







				def __hash__(					self  )  ->   List[str]:
											return hash(self.x  )




def   A   (   _SCREAMING_SNAKE_CASE				,)			->				list[Point]:
							lowerCamelCase     :   list[Point]    				=      []
							if list_of_tuples:
														for p in list_of_tuples:
																					if isinstance(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					):
																												points.append(_SCREAMING_SNAKE_CASE					)
																					else:
																												try:
																																			points.append(Point(p[0]				,p[1]					)					)
																												except (IndexError, TypeError):
																																			print(
																																			    f'''Ignoring deformed point {p}. All points'''
																																			    " must have at least 2 coordinates."					)
							return points

def   A   (   _SCREAMING_SNAKE_CASE					)			->				list[Point]:

							if not hasattr(_SCREAMING_SNAKE_CASE				,"__iter__"					):
														lowerCamelCase     :   str    				=      f'''Expecting an iterable object but got an non-iterable type {points}'''
														raise ValueError(_SCREAMING_SNAKE_CASE					)

							if not points:
														lowerCamelCase     :   Any    				=      f'''Expecting a list of points but got {points}'''
														raise ValueError(_SCREAMING_SNAKE_CASE					)

							return _construct_points(_SCREAMING_SNAKE_CASE					)

def   A   (   _SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					)			->				float:
							lowerCamelCase     :   Any    				=      (a.x * b.y + b.x * c.y + c.x * a.y) - (a.y * b.x + b.y * c.x + c.y * a.x)
							return det

def   A   (   _SCREAMING_SNAKE_CASE					)			->				list[Point]:
							lowerCamelCase     :   Dict    				=      sorted(_validate_input(_SCREAMING_SNAKE_CASE					)					)
							lowerCamelCase     :   Optional[Any]    				=      len(_SCREAMING_SNAKE_CASE					)
							lowerCamelCase     :   Dict    				=      set()

							for i in range(n - 1					):
														for j in range(i + 1				,_SCREAMING_SNAKE_CASE					):
																					lowerCamelCase     :   Optional[int]    				=      False
																					lowerCamelCase     :   Optional[int]    				=      True
																					for k in range(_SCREAMING_SNAKE_CASE					):
																												if k != i and k != j:
																																			lowerCamelCase     :   List[str]    				=      _det(points[i]				,points[j]				,points[k]					)

																																			if det_k > 0:
																																										lowerCamelCase     :   Union[str, Any]    				=      True
																																			elif det_k < 0:
																																										lowerCamelCase     :   str    				=      True
																																			else:
																																										# point[i], point[j], point[k] all lie on a straight line
																																										# if point[k] is to the left of point[i] or it's to the
																																										# right of point[j], then point[i], point[j] cannot be
																																										# part of the convex hull of A
																																										if points[k] < points[i] or points[k] > points[j]:
																																																	lowerCamelCase     :   str    				=      False
																																																	break

																												if points_left_of_ij and points_right_of_ij:
																																			lowerCamelCase     :   List[str]    				=      False
																																			break

																					if ij_part_of_convex_hull:
																												convex_set.update([points[i], points[j]]					)

							return sorted(_SCREAMING_SNAKE_CASE					)

def   A   (   _SCREAMING_SNAKE_CASE					)			->				list[Point]:
							lowerCamelCase     :   Any    				=      sorted(_validate_input(_SCREAMING_SNAKE_CASE					)					)
							lowerCamelCase     :   List[str]    				=      len(_SCREAMING_SNAKE_CASE					)

							# divide all the points into an upper hull and a lower hull
							# the left most point and the right most point are definitely
							# members of the convex hull by definition.
							# use these two anchors to divide all the points into two hulls,
							# an upper hull and a lower hull.

							# all points to the left (above) the line joining the extreme points belong to the
							# upper hull
							# all points to the right (below) the line joining the extreme points below to the
							# lower hull
							# ignore all points on the line joining the extreme points since they cannot be
							# part of the convex hull

							lowerCamelCase     :   int    				=      points[0]
							lowerCamelCase     :   List[str]    				=      points[n - 1]

							lowerCamelCase     :   List[str]    				=      {left_most_point, right_most_point}
							lowerCamelCase     :   Union[str, Any]    				=      []
							lowerCamelCase     :   Optional[int]    				=      []

							for i in range(1				,n - 1					):
														lowerCamelCase     :   List[Any]    				=      _det(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,points[i]					)

														if det > 0:
																					upper_hull.append(points[i]					)
														elif det < 0:
																					lower_hull.append(points[i]					)

							_construct_hull(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					)
							_construct_hull(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					)

							return sorted(_SCREAMING_SNAKE_CASE					)

def   A   (   _SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					)			->				None:
							if points:
														lowerCamelCase     :   List[str]    				=      None
														lowerCamelCase     :   Union[str, Any]    				=      float("-inf"					)
														lowerCamelCase     :   int    				=      []

														for p in points:
																					lowerCamelCase     :   List[str]    				=      _det(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					)

																					if det > 0:
																												candidate_points.append(_SCREAMING_SNAKE_CASE					)

																												if det > extreme_point_distance:
																																			lowerCamelCase     :   str    				=      det
																																			lowerCamelCase     :   List[str]    				=      p

														if extreme_point:
																					_construct_hull(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					)
																					convex_set.add(_SCREAMING_SNAKE_CASE					)
																					_construct_hull(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					)

def   A   (   _SCREAMING_SNAKE_CASE					)			->				list[Point]:
							lowerCamelCase     :   str    				=      sorted(_validate_input(_SCREAMING_SNAKE_CASE					)					)
							lowerCamelCase     :   Union[str, Any]    				=      len(_SCREAMING_SNAKE_CASE					)

							lowerCamelCase     :   int    				=      points[:2]
							for i in range(2				,_SCREAMING_SNAKE_CASE					):
														lowerCamelCase     :   List[str]    				=      _det(convex_hull[1]				,convex_hull[0]				,points[i]					)
														if det > 0:
																					convex_hull.insert(0				,points[i]					)
																					break
														elif det < 0:
																					convex_hull.append(points[i]					)
																					break
														else:
																					lowerCamelCase     :   Any    				=      points[i]
							i += 1

							for j in range(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					):
														if (
														    _det(convex_hull[0]				,convex_hull[-1]				,points[j]					) > 0
														    and _det(convex_hull[-1]				,convex_hull[0]				,points[1]					) < 0
														):
																					# The point lies within the convex hull
																					continue

														convex_hull.insert(0				,points[j]					)
														convex_hull.append(points[j]					)
														while _det(convex_hull[0]				,convex_hull[1]				,convex_hull[2]					) >= 0:
																					del convex_hull[1]
														while _det(convex_hull[-1]				,convex_hull[-2]				,convex_hull[-3]					) <= 0:
																					del convex_hull[-2]

    # `convex_hull` is contains the convex hull in circular order
							return sorted(convex_hull[1:] if len(_SCREAMING_SNAKE_CASE					) > 3 else convex_hull					)

def   A   (   )			->				int:
							lowerCamelCase     :   Dict    				=      [
							    (0, 3),
							    (2, 2),
							    (1, 1),
							    (2, 1),
							    (3, 0),
							    (0, 0),
							    (3, 3),
							    (2, -1),
							    (2, -4),
							    (1, -3),
							]
							# the convex set of points is
							# [(0, 0), (0, 3), (1, -3), (2, -4), (3, 0), (3, 3)]
							lowerCamelCase     :   Optional[Any]    				=      convex_hull_bf(_SCREAMING_SNAKE_CASE					)

							lowerCamelCase     :   Optional[Any]    				=      convex_hull_recursive(_SCREAMING_SNAKE_CASE					)
							assert results_bf == results_recursive

							lowerCamelCase     :   Optional[Any]    				=      convex_hull_melkman(_SCREAMING_SNAKE_CASE					)
							assert results_bf == results_melkman

							print(_SCREAMING_SNAKE_CASE					)


if __name__ == "__main__":
							main()












from __future__ import annotations

def   A   (   _SCREAMING_SNAKE_CASE					)			->				list[int]:
       assert (
           isinstance(_SCREAMING_SNAKE_CASE				,_SCREAMING_SNAKE_CASE					) and mask > 0
       ), f'''mask needs to be positive integer, your input {mask}'''
       lowerCamelCase     :   str    				=      []
       lowerCamelCase     :   Any    				=      mask

       while submask:
              all_submasks.append(_SCREAMING_SNAKE_CASE					)
              lowerCamelCase     :   Union[str, Any]    				=      (submask - 1) & mask

       return all_submasks


if __name__ == "__main__":
       import doctest

       doctest.testmod()






