






"""simple docstring"""





from __future__ import annotations

from collections.abc import Iterable







class 		_a       :





			'''simple docstring'''






			def __init__(   self			,				UpperCAmelCase_			,				UpperCAmelCase_)	-> Optional[Any]:




										'''simple docstring'''




										lowercase__:    str        =      float(UpperCAmelCase_), float(UpperCAmelCase_)





			def __eq__(   self			,				UpperCAmelCase_)	-> Dict:




										'''simple docstring'''




										return self.x == other.x and self.y == other.y





			def __ne__(   self			,				UpperCAmelCase_)	-> Union[str, Any]:




										'''simple docstring'''




										return not self == other





			def __gt__(   self			,				UpperCAmelCase_)	-> Dict:




										'''simple docstring'''




										if self.x > other.x:
																	return True
										elif self.x == other.x:
																	return self.y > other.y
										return False





			def __lt__(   self			,				UpperCAmelCase_)	-> Optional[int]:




										'''simple docstring'''




										return not self > other





			def __ge__(   self			,				UpperCAmelCase_)	-> Union[str, Any]:




										'''simple docstring'''




										if self.x > other.x:
																	return True
										elif self.x == other.x:
																	return self.y >= other.y
										return False





			def __le__(   self			,				UpperCAmelCase_)	-> List[Any]:




										'''simple docstring'''




										if self.x < other.x:
																	return True
										elif self.x == other.x:
																	return self.y <= other.y
										return False





			def __repr__(   self)	-> List[Any]:




										'''simple docstring'''




										return F"""({self.x}, {self.y})"""





			def __hash__(   self)	-> Any:




										'''simple docstring'''




										return hash(self.x)







def     A(  snake_case_						,       ):
							"""simple docstring"""





							lowercase__:    list[Point]        =      []
							if list_of_tuples:
														for p in list_of_tuples:
																					if isinstance(snake_case_						,       snake_case_		):
																												points.append(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(  snake_case_		):
							"""simple docstring"""





							if not hasattr(snake_case_						,       "__iter__"		):
														lowercase__:    Tuple        =      F"""Expecting an iterable object but got an non-iterable type {points}"""
														raise ValueError(snake_case_		)

							if not points:
														lowercase__:    Dict        =      F"""Expecting a list of points but got {points}"""
														raise ValueError(snake_case_		)

							return _construct_points(snake_case_		)






def     A(  snake_case_						,       snake_case_						,       snake_case_		):
							"""simple docstring"""





							lowercase__:    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(  snake_case_		):
							"""simple docstring"""





							lowercase__:    Optional[int]        =      sorted(_validate_input(snake_case_		)		)
							lowercase__:    int        =      len(snake_case_		)
							lowercase__:    Optional[int]        =      set()

							for i in range(n - 1		):
														for j in range(i + 1						,       snake_case_		):
																					lowercase__:    Dict        =      False
																					lowercase__:    List[str]        =      True
																					for k in range(snake_case_		):
																												if k != i and k != j:
																																			lowercase__:    List[Any]        =      _det(points[i]						,       points[j]						,       points[k]		)

																																			if det_k > 0:
																																										lowercase__:    List[Any]        =      True
																																			elif det_k < 0:
																																										lowercase__:    Any        =      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]:
																																																	lowercase__:    Any        =      False
																																																	break

																												if points_left_of_ij and points_right_of_ij:
																																			lowercase__:    Dict        =      False
																																			break

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

							return sorted(snake_case_		)






def     A(  snake_case_		):
							"""simple docstring"""





							lowercase__:    Any        =      sorted(_validate_input(snake_case_		)		)
							lowercase__:    str        =      len(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

							lowercase__:    int        =      points[0]
							lowercase__:    int        =      points[n - 1]

							lowercase__:    Tuple        =      {left_most_point, right_most_point}
							lowercase__:    List[str]        =      []
							lowercase__:    List[Any]        =      []

							for i in range(1						,       n - 1		):
														lowercase__:    str        =      _det(snake_case_						,       snake_case_						,       points[i]		)

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

							_construct_hull(snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		)
							_construct_hull(snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		)

							return sorted(snake_case_		)






def     A(  snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		):
							"""simple docstring"""





							if points:
														lowercase__:    str        =      None
														lowercase__:    int        =      float("-inf"		)
														lowercase__:    Union[str, Any]        =      []

														for p in points:
																					lowercase__:    Any        =      _det(snake_case_						,       snake_case_						,       snake_case_		)

																					if det > 0:
																												candidate_points.append(snake_case_		)

																												if det > extreme_point_distance:
																																			lowercase__:    Dict        =      det
																																			lowercase__:    List[str]        =      p

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






def     A(  snake_case_		):
							"""simple docstring"""





							lowercase__:    Optional[Any]        =      sorted(_validate_input(snake_case_		)		)
							lowercase__:    List[str]        =      len(snake_case_		)

							lowercase__:    Union[str, Any]        =      points[:2]
							for i in range(2						,       snake_case_		):
														lowercase__:    List[Any]        =      _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:
																					lowercase__:    List[Any]        =      points[i]
							i += 1

							for j in range(snake_case_						,       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(snake_case_		) > 3 else convex_hull		)






def     A(  ):
							"""simple docstring"""





							lowercase__:    List[str]        =      [
							    (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)]
							lowercase__:    List[str]        =      convex_hull_bf(snake_case_		)

							lowercase__:    Union[str, Any]        =      convex_hull_recursive(snake_case_		)
							assert results_bf == results_recursive

							lowercase__:    Optional[Any]        =      convex_hull_melkman(snake_case_		)
							assert results_bf == results_melkman

							print(snake_case_		)


if __name__ == "__main__":
		main()














"""simple docstring"""





UpperCamelCase			=      {
    """0""": 1,
    """1""": 1,
    """2""": 2,
    """3""": 6,
    """4""": 24,
    """5""": 1_20,
    """6""": 7_20,
    """7""": 50_40,
    """8""": 4_03_20,
    """9""": 36_28_80,
}

UpperCamelCase			=      {1_45: 1_45}

UpperCamelCase			=      {
    1_45: 0,
    1_69: 3,
    3_63_01: 3,
    14_54: 3,
    8_71: 2,
    4_53_61: 2,
    8_72: 2,
}






def     A(  snake_case_		):
       """simple docstring"""





       if n in CACHE_SUM_DIGIT_FACTORIALS:
              return CACHE_SUM_DIGIT_FACTORIALS[n]
       lowercase__:    Any        =      sum(DIGIT_FACTORIALS[let] for let in str(snake_case_		)		)
       lowercase__:    Optional[Any]        =      ret
       return ret






def     A(  snake_case_						,       snake_case_ = None		):
       """simple docstring"""





       lowercase__:    Optional[int]        =      previous or set()
       if n in CHAIN_LENGTH_CACHE:
              return CHAIN_LENGTH_CACHE[n]
       lowercase__:    Tuple        =      sum_digit_factorials(snake_case_		)
       if next_number in previous:
              lowercase__:    List[Any]        =      0
              return 0
       else:
              previous.add(snake_case_		)
              lowercase__:    Optional[int]        =      1 + chain_length(snake_case_						,       snake_case_		)
              lowercase__:    Any        =      ret
              return ret






def     A(  snake_case_ = 60						,       snake_case_ = 1000000		):
       """simple docstring"""





       return sum(1 for i in range(1						,       snake_case_		) if chain_length(snake_case_		) == num_terms		)


if __name__ == "__main__":
  print(F"{solution() = }")






