




def        __SCREAMING_SNAKE_CASE (	lowercase_   )							->						list[int]:

    """simple docstring"""


    if len(lowercase_   ) > 1:
        __UpperCamelCase			=    len(lowercase_   ) // 2  # Finds the middle of the array
        __UpperCamelCase			=    arr[
            :middle_length
        ]  # Creates an array of the elements in the first half.
        __UpperCamelCase			=    arr[
            middle_length:
        ]  # Creates an array of the elements in the second half.
        __UpperCamelCase			=    len(lowercase_   )
        __UpperCamelCase			=    len(lowercase_   )
        merge(lowercase_   )  # Starts sorting the left.
        merge(lowercase_   )  # Starts sorting the right
        __UpperCamelCase			=    0  # Left Counter
        __UpperCamelCase			=    0  # Right Counter
        __UpperCamelCase			=    0  # Position Counter
        while (
            left_index < left_size and right_index < right_size
        ):  # Runs until the lowers size of the left and right are sorted.
            if left_array[left_index] < right_array[right_index]:
                __UpperCamelCase			=    left_array[left_index]
                left_index += 1
            else:
                __UpperCamelCase			=    right_array[right_index]
                right_index += 1
            index += 1
        while (
            left_index < left_size
        ):  # Adds the left over elements in the left half of the array
            __UpperCamelCase			=    left_array[left_index]
            left_index += 1
            index += 1
        while (
            right_index < right_size
        ):  # Adds the left over elements in the right half of the array
            __UpperCamelCase			=    right_array[right_index]
            right_index += 1
            index += 1
    return arr


if __name__ == "__main__":
 import doctest

 doctest.testmod()







from __future__ import annotations

from collections.abc import Iterable
class 							a_					:






 '''simple docstring'''






 def __init__(     self   :    int     ,       lowercase__   :    Optional[int]     ,       lowercase__   :    List[str]):







        '''simple docstring'''



        lowerCAmelCase__      ,   lowerCAmelCase__	      =   float(lowercase__), float(lowercase__)




 def __eq__(     self   :    int     ,       lowercase__   :    int):







        '''simple docstring'''



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




 def __ne__(     self   :    Tuple     ,       lowercase__   :    List[Any]):







        '''simple docstring'''



        return not self == other




 def __gt__(     self   :    Optional[Any]     ,       lowercase__   :    List[Any]):







        '''simple docstring'''



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




 def __lt__(     self   :    str     ,       lowercase__   :    List[str]):







        '''simple docstring'''



        return not self > other




 def __ge__(     self   :    List[Any]     ,       lowercase__   :    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   :    Dict     ,       lowercase__   :    str):







        '''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[str]):







        '''simple docstring'''



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




 def __hash__(     self   :    Dict):







        '''simple docstring'''



        return hash(self.x)






def  __lowerCamelCase						(		lowerCAmelCase__      ,    ):
       lowerCAmelCase__	      =   []
       if list_of_tuples:
              for p in list_of_tuples:
                     if isinstance(lowerCAmelCase__      ,    lowerCAmelCase__      ):
                            points.append(lowerCAmelCase__      )
                     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  __lowerCamelCase						(		lowerCAmelCase__      ):

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

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

       return _construct_points(lowerCAmelCase__      )






def  __lowerCamelCase						(		lowerCAmelCase__      ,    lowerCAmelCase__      ,    lowerCAmelCase__      ):
       lowerCAmelCase__	      =   (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  __lowerCamelCase						(		lowerCAmelCase__      ):
       lowerCAmelCase__	      =   sorted(_validate_input(lowerCAmelCase__      )      )
       lowerCAmelCase__	      =   len(lowerCAmelCase__      )
       lowerCAmelCase__	      =   set()

       for i in range(n - 1      ):
              for j in range(i + 1      ,    lowerCAmelCase__      ):
                     lowerCAmelCase__	      =   lowerCAmelCase__	      =   False
                     lowerCAmelCase__	      =   True
                     for k in range(lowerCAmelCase__      ):
                            if k != i and k != j:
                                   lowerCAmelCase__	      =   _det(points[i]      ,    points[j]      ,    points[k]      )

                                   if det_k > 0:
                                          lowerCAmelCase__	      =   True
                                   elif det_k < 0:
                                          lowerCAmelCase__	      =   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]:
                                                 lowerCAmelCase__	      =   False
                                                 break

                            if points_left_of_ij and points_right_of_ij:
                                   lowerCAmelCase__	      =   False
                                   break

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

       return sorted(lowerCAmelCase__      )






def  __lowerCamelCase						(		lowerCAmelCase__      ):
       lowerCAmelCase__	      =   sorted(_validate_input(lowerCAmelCase__      )      )
       lowerCAmelCase__	      =   len(lowerCAmelCase__      )

       # 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

       lowerCAmelCase__	      =   points[0]
       lowerCAmelCase__	      =   points[n - 1]

       lowerCAmelCase__	      =   {left_most_point, right_most_point}
       lowerCAmelCase__	      =   []
       lowerCAmelCase__	      =   []

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

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

       _construct_hull(lowerCAmelCase__      ,    lowerCAmelCase__      ,    lowerCAmelCase__      ,    lowerCAmelCase__      )
       _construct_hull(lowerCAmelCase__      ,    lowerCAmelCase__      ,    lowerCAmelCase__      ,    lowerCAmelCase__      )

       return sorted(lowerCAmelCase__      )






def  __lowerCamelCase						(		lowerCAmelCase__      ,    lowerCAmelCase__      ,    lowerCAmelCase__      ,    lowerCAmelCase__      ):
       if points:
              lowerCAmelCase__	      =   None
              lowerCAmelCase__	      =   float('-inf'      )
              lowerCAmelCase__	      =   []

              for p in points:
                     lowerCAmelCase__	      =   _det(lowerCAmelCase__      ,    lowerCAmelCase__      ,    lowerCAmelCase__      )

                     if det > 0:
                            candidate_points.append(lowerCAmelCase__      )

                            if det > extreme_point_distance:
                                   lowerCAmelCase__	      =   det
                                   lowerCAmelCase__	      =   p

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






def  __lowerCamelCase						(		lowerCAmelCase__      ):
       lowerCAmelCase__	      =   sorted(_validate_input(lowerCAmelCase__      )      )
       lowerCAmelCase__	      =   len(lowerCAmelCase__      )

       lowerCAmelCase__	      =   points[:2]
       for i in range(2      ,    lowerCAmelCase__      ):
              lowerCAmelCase__	      =   _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:
                     lowerCAmelCase__	      =   points[i]
       i += 1

       for j in range(lowerCAmelCase__      ,    lowerCAmelCase__      ):
              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(lowerCAmelCase__      ) > 3 else convex_hull      )






def  __lowerCamelCase						(		):
       lowerCAmelCase__	      =   [
           (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)]
       lowerCAmelCase__	      =   convex_hull_bf(lowerCAmelCase__      )

       lowerCAmelCase__	      =   convex_hull_recursive(lowerCAmelCase__      )
       assert results_bf == results_recursive

       lowerCAmelCase__	      =   convex_hull_melkman(lowerCAmelCase__      )
       assert results_bf == results_melkman

       print(lowerCAmelCase__      )


if __name__ == "__main__":
    main()
