"""simple docstring"""




from __future__ import annotations

from collections.abc import Iterable
class      UpperCamelCase__:





						def __init__(							self		,__UpperCAmelCase		,__UpperCAmelCase  )       ->					Any:
													A__  ,	A__						  =				float(__UpperCAmelCase  ), float(__UpperCAmelCase  )





						def __eq__(							self		,__UpperCAmelCase  )       ->					str:
													return self.x == other.x and self.y == other.y





						def __ne__(							self		,__UpperCAmelCase  )       ->					List[str]:
													return not self == other





						def __gt__(							self		,__UpperCAmelCase  )       ->					Tuple:
													if self.x > other.x:
																				return True
													elif self.x == other.x:
																				return self.y > other.y
													return False





						def __lt__(							self		,__UpperCAmelCase  )       ->					List[Any]:
													return not self > other





						def __ge__(							self		,__UpperCAmelCase  )       ->					str:
													if self.x > other.x:
																				return True
													elif self.x == other.x:
																				return self.y >= other.y
													return False





						def __le__(							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 __repr__(							self  )       ->					Dict:
													return f'''({self.x}, {self.y})'''





						def __hash__(							self  )       ->					Tuple:
													return hash(self.x  )






def 	UpperCAmelCase       (						UpperCamelCase__							,     ):






							"""simple docstring"""





							A__						  =				[]
							if list_of_tuples:
														for p in list_of_tuples:
																					if isinstance(UpperCamelCase__							,     UpperCamelCase__					):
																												points.append(UpperCamelCase__					)
																					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 	UpperCAmelCase       (						UpperCamelCase__					):






							"""simple docstring"""





							if not hasattr(UpperCamelCase__							,     '__iter__'					):
														A__						  =				F'''Expecting an iterable object but got an non-iterable type {points}'''
														raise ValueError(UpperCamelCase__					)

							if not points:
														A__						  =				F'''Expecting a list of points but got {points}'''
														raise ValueError(UpperCamelCase__					)

							return _construct_points(UpperCamelCase__					)






def 	UpperCAmelCase       (						UpperCamelCase__							,     UpperCamelCase__							,     UpperCamelCase__					):






							"""simple docstring"""





							A__						  =				(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 	UpperCAmelCase       (						UpperCamelCase__					):






							"""simple docstring"""





							A__						  =				sorted(_validate_input(UpperCamelCase__					)					)
							A__						  =				len(UpperCamelCase__					)
							A__						  =				set()

							for i in range(n - 1					):
														for j in range(i + 1							,     UpperCamelCase__					):
																					A__						  =				A__						  =				False
																					A__						  =				True
																					for k in range(UpperCamelCase__					):
																												if k != i and k != j:
																																			A__						  =				_det(points[i]							,     points[j]							,     points[k]					)

																																			if det_k > 0:
																																										A__						  =				True
																																			elif det_k < 0:
																																										A__						  =				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]:
																																																	A__						  =				False
																																																	break

																												if points_left_of_ij and points_right_of_ij:
																																			A__						  =				False
																																			break

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

							return sorted(UpperCamelCase__					)






def 	UpperCAmelCase       (						UpperCamelCase__					):






							"""simple docstring"""





							A__						  =				sorted(_validate_input(UpperCamelCase__					)					)
							A__						  =				len(UpperCamelCase__					)

							# 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

							A__						  =				points[0]
							A__						  =				points[n - 1]

							A__						  =				{left_most_point, right_most_point}
							A__						  =				[]
							A__						  =				[]

							for i in range(1							,     n - 1					):
														A__						  =				_det(UpperCamelCase__							,     UpperCamelCase__							,     points[i]					)

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

							_construct_hull(UpperCamelCase__							,     UpperCamelCase__							,     UpperCamelCase__							,     UpperCamelCase__					)
							_construct_hull(UpperCamelCase__							,     UpperCamelCase__							,     UpperCamelCase__							,     UpperCamelCase__					)

							return sorted(UpperCamelCase__					)






def 	UpperCAmelCase       (						UpperCamelCase__							,     UpperCamelCase__							,     UpperCamelCase__							,     UpperCamelCase__					):






							"""simple docstring"""





							if points:
														A__						  =				None
														A__						  =				float('-inf'					)
														A__						  =				[]

														for p in points:
																					A__						  =				_det(UpperCamelCase__							,     UpperCamelCase__							,     UpperCamelCase__					)

																					if det > 0:
																												candidate_points.append(UpperCamelCase__					)

																												if det > extreme_point_distance:
																																			A__						  =				det
																																			A__						  =				p

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






def 	UpperCAmelCase       (						UpperCamelCase__					):






							"""simple docstring"""





							A__						  =				sorted(_validate_input(UpperCamelCase__					)					)
							A__						  =				len(UpperCamelCase__					)

							A__						  =				points[:2]
							for i in range(2							,     UpperCamelCase__					):
														A__						  =				_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:
																					A__						  =				points[i]
							i += 1

							for j in range(UpperCamelCase__							,     UpperCamelCase__					):
														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(UpperCamelCase__					) > 3 else convex_hull					)






def 	UpperCAmelCase       (						):






							"""simple docstring"""





							A__						  =				[
							    (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)]
							A__						  =				convex_hull_bf(UpperCamelCase__					)

							A__						  =				convex_hull_recursive(UpperCamelCase__					)
							assert results_bf == results_recursive

							A__						  =				convex_hull_melkman(UpperCamelCase__					)
							assert results_bf == results_melkman

							print(UpperCamelCase__					)


if __name__ == "__main__":
				main()






"""simple docstring"""




def 	UpperCAmelCase       (						UpperCamelCase__ = 100							,     UpperCamelCase__ = 100					):






							"""simple docstring"""





							return max(
							    sum(int(UpperCamelCase__					) for x in str(base**power					)					)
							    for base in range(UpperCamelCase__					)
							    for power in range(UpperCamelCase__					)					)


# Tests
if __name__ == "__main__":
				import doctest

				doctest.testmod()





