


from __future__ import annotations

from collections.abc import Iterable




class 			lowercase_				:





				def __init__(							self					,     lowercase_					,     lowercase_)     ->   Optional[Any]:
											a__						,    a__     						=float(lowercase_), float(lowercase_)





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





				def __ne__(							self					,     lowercase_)     ->   str:
											return not self == other





				def __gt__(							self					,     lowercase_)     ->   List[str]:
											if self.x > other.x:
																		return True
											elif self.x == other.x:
																		return self.y > other.y
											return False





				def __lt__(							self					,     lowercase_)     ->   int:
											return not self > other





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





				def __repr__(							self)     ->   Optional[int]:
											return F"""({self.x}, {self.y})"""





				def __hash__(							self)     ->   Optional[Any]:
											return hash(self.x)







def      _lowercase(				__a							:	list[Point] | list[list[float]] | Iterable[list[float]]						,    ):
							a__     						=[]
							if list_of_tuples:
														for p in list_of_tuples:
																					if isinstance(__a						,    __a    ):
																												points.append(__a    )
																					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      _lowercase(				__a							:	list[Point] | list[list[float]]    ):

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

							if not points:
														a__     						=f"""Expecting a list of points but got {points}"""
														raise ValueError(__a    )

							return _construct_points(__a    )






def      _lowercase(				__a							:	Point						,    __a							:	Point						,    __a							:	Point    ):
							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      _lowercase(				__a							:	list[Point]    ):
							a__     						=sorted(_validate_input(__a    )    )
							a__     						=len(__a    )
							a__     						=set()

							for i in range(n - 1    ):
														for j in range(i + 1						,    __a    ):
																					a__     						=a__     						=False
																					a__     						=True
																					for k in range(__a    ):
																												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(__a    )






def      _lowercase(				__a							:	list[Point]    ):
							a__     						=sorted(_validate_input(__a    )    )
							a__     						=len(__a    )

							# 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(__a						,    __a						,    points[i]    )

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

							_construct_hull(__a						,    __a						,    __a						,    __a    )
							_construct_hull(__a						,    __a						,    __a						,    __a    )

							return sorted(__a    )






def      _lowercase(				__a							:	list[Point]						,    __a							:	Point						,    __a							:	Point						,    __a							:	set[Point]    ):
							if points:
														a__     						=None
														a__     						=float('-inf'    )
														a__     						=[]

														for p in points:
																					a__     						=_det(__a						,    __a						,    __a    )

																					if det > 0:
																												candidate_points.append(__a    )

																												if det > extreme_point_distance:
																																			a__     						=det
																																			a__     						=p

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






def      _lowercase(				__a							:	list[Point]    ):
							a__     						=sorted(_validate_input(__a    )    )
							a__     						=len(__a    )

							a__     						=points[:2]
							for i in range(2						,    __a    ):
														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(__a						,    __a    ):
														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(__a    ) > 3 else convex_hull    )






def      _lowercase(				):
							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(__a    )

							a__     						=convex_hull_recursive(__a    )
							assert results_bf == results_recursive

							a__     						=convex_hull_melkman(__a    )
							assert results_bf == results_melkman

							print(__a    )


if __name__ == "__main__":
	main()





from __future__ import annotations






def      _lowercase(				__a							:	float | int						,    __a							:	float | int						,    __a							:	float | int						,    ):
							if not all((nth_term, start_term_a, common_ratio_r)    ):
														return []
							a__     						=[]
							a__     						=1
							a__     						=common_ratio_r
							for _ in range(int(__a    )    ):
														if not series:
																					series.append(__a    )
														else:
																					power += 1
																					series.append(float(start_term_a * multiple    )    )
																					a__     						=pow(float(__a    )						,    __a    )
							return series


if __name__ == "__main__":
	import doctest

	doctest.testmod()

	_lowerCAmelCase:	Tuple      							=					float(input('Enter the last number (n term) of the Geometric Series'))
	_lowerCAmelCase:	Optional[int]      							=					float(input('Enter the starting term (a) of the Geometric Series'))
	_lowerCAmelCase:	List[Any]      							=					float(
	    input('Enter the common ratio between two terms (r) of the Geometric Series')
	)
	print('Formula of Geometric Series => a + ar + ar^2 ... +ar^n')
	print(geometric_series(nth_term, start_term_a, common_ratio_r))

