





"""simple docstring"""



from __future__ import annotations

from collections.abc import Iterable






class 	__snake_case      :





	"""simple docstring"""


	def __init__(					self      ,		__lowerCamelCase      ,		__lowerCamelCase						):



					'''simple docstring'''




					__A					:		List[Any]						=  float(__lowerCamelCase						), float(__lowerCamelCase						)

	def __eq__(					self      ,		__lowerCamelCase						):



					'''simple docstring'''




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

	def __ne__(					self      ,		__lowerCamelCase						):



					'''simple docstring'''




					return not self == other

	def __gt__(					self      ,		__lowerCamelCase						):



					'''simple docstring'''




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

	def __lt__(					self      ,		__lowerCamelCase						):



					'''simple docstring'''




					return not self > other

	def __ge__(					self      ,		__lowerCamelCase						):



					'''simple docstring'''




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

	def __le__(					self      ,		__lowerCamelCase						):



					'''simple docstring'''




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

	def __repr__(					self						):



					'''simple docstring'''




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







	def __hash__(					self						):



					'''simple docstring'''




					return hash(self.x						)







def 	__lowercase  (     snake_case_						:						list[Point] | list[list[float]] | Iterable[list[float]]       ,)     ->list[Point]:



				'''simple docstring'''







				__A					:		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 	__lowercase  (     snake_case_						:						list[Point] | list[list[float]]		)     ->list[Point]:



				'''simple docstring'''







				if not hasattr(snake_case_       ,'''__iter__'''		):
								__A					:		Union[str, Any]						=  F"""Expecting an iterable object but got an non-iterable type {points}"""
								raise ValueError(snake_case_		)

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

				return _construct_points(snake_case_		)







def 	__lowercase  (     snake_case_						:						Point       ,snake_case_						:						Point       ,snake_case_						:						Point		)     ->float:



				'''simple docstring'''







				__A					:		List[str]						=  (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  (     snake_case_						:						list[Point]		)     ->list[Point]:



				'''simple docstring'''







				__A					:		Tuple						=  sorted(_validate_input(snake_case_		)		)
				__A					:		Any						=  len(snake_case_		)
				__A					:		Union[str, Any]						=  set()

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

																				if det_k > 0:
																								__A					:		Optional[int]						=  True
																				elif det_k < 0:
																								__A					:		List[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]:
																												__A					:		Any						=  False
																												break

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

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

				return sorted(snake_case_		)







def 	__lowercase  (     snake_case_						:						list[Point]		)     ->list[Point]:



				'''simple docstring'''







				__A					:		int						=  sorted(_validate_input(snake_case_		)		)
				__A					:		List[Any]						=  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

				__A					:		Tuple						=  points[0]
				__A					:		str						=  points[n - 1]

				__A					:		str						=  {left_most_point, right_most_point}
				__A					:		List[str]						=  []
				__A					:		Dict						=  []

				for i in range(1       ,n - 1		):
								__A					:		List[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 	__lowercase  (     snake_case_						:						list[Point]       ,snake_case_						:						Point       ,snake_case_						:						Point       ,snake_case_						:						set[Point]		)     ->None:



				'''simple docstring'''







				if points:
								__A					:		List[str]						=  None
								__A					:		Dict						=  float('''-inf'''		)
								__A					:		Optional[Any]						=  []

								for p in points:
												__A					:		List[Any]						=  _det(snake_case_       ,snake_case_       ,snake_case_		)

												if det > 0:
																candidate_points.append(snake_case_		)

																if det > extreme_point_distance:
																				__A					:		str						=  det
																				__A					:		Dict						=  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 	__lowercase  (     snake_case_						:						list[Point]		)     ->list[Point]:



				'''simple docstring'''







				__A					:		Union[str, Any]						=  sorted(_validate_input(snake_case_		)		)
				__A					:		Dict						=  len(snake_case_		)

				__A					:		Any						=  points[:2]
				for i in range(2       ,snake_case_		):
								__A					:		Tuple						=  _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					:		Tuple						=  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 	__lowercase  (     )     ->Optional[Any]:



				'''simple docstring'''







				__A					:		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)]
				__A					:		Optional[int]						=  convex_hull_bf(snake_case_		)

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

				__A					:		Tuple						=  convex_hull_melkman(snake_case_		)
				assert results_bf == results_melkman

				print(snake_case_		)


if __name__ == "__main__":
							main()











"""simple docstring"""



import decimal







def 	__lowercase  (     )     ->int:



    '''simple docstring'''







    __A					:		List[str]						=  0
    __A					:		Tuple						=  decimal.Context(prec=105		)
    for i in range(2       ,100		):
        __A					:		int						=  decimal.Decimal(snake_case_		)
        __A					:		str						=  number.sqrt(snake_case_		)
        if len(str(snake_case_		)		) > 1:
            answer += int(str(snake_case_		)[0]		)
            __A					:		Optional[Any]						=  str(snake_case_		)[2:101]
            answer += sum(int(snake_case_		) for x in sqrt_number_str		)
    return answer


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       print(f'''{solution() = }''')




