





"""simple docstring"""



from __future__ import annotations

from collections.abc import Iterable






class 	__snake_case      :





 """simple docstring"""


 def __init__(					self      ,		__lowerCamelCase      ,		__lowerCamelCase						):



     '''simple docstring'''




     __A				, __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() = }''')




