




"""simple docstring"""




from __future__ import annotations

from collections.abc import Iterable

class        UpperCAmelCase_  :
		def __init__(		self							,		UpperCamelCase_							,		UpperCamelCase_   )		->   Union[str, Any]:
					__lowercase	:       Optional[int]				=   float(SCREAMING_SNAKE_CASE__   ), float(SCREAMING_SNAKE_CASE__   )
		def __eq__(		self							,		UpperCamelCase_   )		->   Tuple:
					return self.x == other.x and self.y == other.y
		def __ne__(		self							,		UpperCamelCase_   )		->   Any:
					return not self == other
		def __gt__(		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 __lt__(		self							,		UpperCamelCase_   )		->   List[str]:
					return not self > other
		def __ge__(		self							,		UpperCamelCase_   )		->   Dict:
					if self.x > other.x:
								return True
					elif self.x == other.x:
								return self.y >= other.y
					return False
		def __le__(		self							,		UpperCamelCase_   )		->   Optional[int]:
					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   )		->   Dict:
					return hash(self.x   )







def       __UpperCAmelCase      (				__UpperCamelCase					,							):






			"""simple docstring"""




			__lowercase	:       list[Point]				=   []
			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__'''					):
						__lowercase	:       List[Any]				=   f"""Expecting an iterable object but got an non-iterable type {points}"""
						raise ValueError(__UpperCamelCase					)

			if not points:
						__lowercase	:       Tuple				=   f"""Expecting a list of points but got {points}"""
						raise ValueError(__UpperCamelCase					)

			return _construct_points(__UpperCamelCase					)







def       __UpperCAmelCase      (				__UpperCamelCase					,							__UpperCamelCase					,							__UpperCamelCase					):






			"""simple docstring"""




			__lowercase	:       Optional[int]				=   (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"""




			__lowercase	:       int				=   sorted(_validate_input(__UpperCamelCase					)					)
			__lowercase	:       int				=   len(__UpperCamelCase					)
			__lowercase	:       Union[str, Any]				=   set()

			for i in range(n - 1					):
						for j in range(i + 1					,							__UpperCamelCase					):
									__lowercase	:       List[str]				=   False
									__lowercase	:       Optional[Any]				=   True
									for k in range(__UpperCamelCase					):
												if k != i and k != j:
															__lowercase	:       Union[str, Any]				=   _det(points[i]					,							points[j]					,							points[k]					)

															if det_k > 0:
																		__lowercase	:       Union[str, Any]				=   True
															elif det_k < 0:
																		__lowercase	:       int				=   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	:       Dict				=   False
																					break

												if points_left_of_ij and points_right_of_ij:
															__lowercase	:       str				=   False
															break

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

			return sorted(__UpperCamelCase					)







def       __UpperCAmelCase      (				__UpperCamelCase					):






			"""simple docstring"""




			__lowercase	:       List[Any]				=   sorted(_validate_input(__UpperCamelCase					)					)
			__lowercase	:       Any				=   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

			__lowercase	:       Dict				=   points[0]
			__lowercase	:       int				=   points[n - 1]

			__lowercase	:       Tuple				=   {left_most_point, right_most_point}
			__lowercase	:       Optional[Any]				=   []
			__lowercase	:       Optional[Any]				=   []

			for i in range(1					,							n - 1					):
						__lowercase	:       Dict				=   _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:
						__lowercase	:       int				=   None
						__lowercase	:       Any				=   float('''-inf'''					)
						__lowercase	:       Union[str, Any]				=   []

						for p in points:
									__lowercase	:       List[str]				=   _det(__UpperCamelCase					,							__UpperCamelCase					,							__UpperCamelCase					)

									if det > 0:
												candidate_points.append(__UpperCamelCase					)

												if det > extreme_point_distance:
															__lowercase	:       Tuple				=   det
															__lowercase	:       List[str]				=   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"""




			__lowercase	:       Union[str, Any]				=   sorted(_validate_input(__UpperCamelCase					)					)
			__lowercase	:       Optional[Any]				=   len(__UpperCamelCase					)

			__lowercase	:       Optional[Any]				=   points[:2]
			for i in range(2					,							__UpperCamelCase					):
						__lowercase	:       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:
									__lowercase	:       Any				=   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"""




			__lowercase	:       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	:       Dict				=   convex_hull_bf(__UpperCamelCase					)

			__lowercase	:       str				=   convex_hull_recursive(__UpperCamelCase					)
			assert results_bf == results_recursive

			__lowercase	:       List[Any]				=   convex_hull_melkman(__UpperCamelCase					)
			assert results_bf == results_melkman

			print(__UpperCamelCase					)


if __name__ == "__main__":
			main()









"""simple docstring"""




from collections import defaultdict







def       __UpperCAmelCase      (				__UpperCamelCase					):
			__lowercase	:       defaultdict[str, int]				=   defaultdict(__UpperCamelCase					)
			# Creating a dictionary containing count of each word
			for word in sentence.split():
						occurrence[word] += 1
			return occurrence


if __name__ == "__main__":
			for word, count in word_occurrence('INPUT STRING').items():
						print(F"{word}: {count}")



