


from __future__ import annotations

from collections.abc import Iterable
class      UpperCAmelCase_ :

   '''simple docstring'''




   def __init__(		self						,    __A						,    __A			):






     """simple docstring"""





     lowerCamelCase   ,     lowerCamelCase			: int   =    float(__A			), float(__A			)






   def __eq__(		self						,    __A			):






     """simple docstring"""





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






   def __ne__(		self						,    __A			):






     """simple docstring"""





     return not self == other






   def __gt__(		self						,    __A			):






     """simple docstring"""





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






   def __lt__(		self						,    __A			):






     """simple docstring"""





     return not self > other






   def __ge__(		self						,    __A			):






     """simple docstring"""





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






   def __le__(		self						,    __A			):






     """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_(  SCREAMING_SNAKE_CASE_				,						):


  '''simple docstring'''

  lowerCamelCase			: list[Point]   =    []
  if list_of_tuples:
    for p in list_of_tuples:
      if isinstance(SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_     ):
        points.append(SCREAMING_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_(  SCREAMING_SNAKE_CASE_     ):


  '''simple docstring'''

  if not hasattr(SCREAMING_SNAKE_CASE_				,						"__iter__"     ):
    lowerCamelCase			: Optional[Any]   =    f"""Expecting an iterable object but got an non-iterable type {points}"""
    raise ValueError(SCREAMING_SNAKE_CASE_     )

  if not points:
    lowerCamelCase			: Union[str, Any]   =    f"""Expecting a list of points but got {points}"""
    raise ValueError(SCREAMING_SNAKE_CASE_     )

  return _construct_points(SCREAMING_SNAKE_CASE_     )






def 		lowercase_(  SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_     ):


  '''simple docstring'''

  lowerCamelCase			: Union[str, Any]   =    (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_(  SCREAMING_SNAKE_CASE_     ):


  '''simple docstring'''

  lowerCamelCase			: Any   =    sorted(_validate_input(SCREAMING_SNAKE_CASE_     )     )
  lowerCamelCase			: List[str]   =    len(SCREAMING_SNAKE_CASE_     )
  lowerCamelCase			: Tuple   =    set()

  for i in range(n - 1     ):
    for j in range(i + 1				,						SCREAMING_SNAKE_CASE_     ):
      lowerCamelCase			: List[Any]   =    False
      lowerCamelCase			: Any   =    True
      for k in range(SCREAMING_SNAKE_CASE_     ):
        if k != i and k != j:
          lowerCamelCase			: Optional[int]   =    _det(points[i]				,						points[j]				,						points[k]     )

          if det_k > 0:
            lowerCamelCase			: List[str]   =    True
          elif det_k < 0:
            lowerCamelCase			: Optional[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]:
              lowerCamelCase			: Dict   =    False
              break

        if points_left_of_ij and points_right_of_ij:
          lowerCamelCase			: Optional[Any]   =    False
          break

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

  return sorted(SCREAMING_SNAKE_CASE_     )






def 		lowercase_(  SCREAMING_SNAKE_CASE_     ):


  '''simple docstring'''

  lowerCamelCase			: List[str]   =    sorted(_validate_input(SCREAMING_SNAKE_CASE_     )     )
  lowerCamelCase			: Any   =    len(SCREAMING_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

  lowerCamelCase			: int   =    points[0]
  lowerCamelCase			: str   =    points[n - 1]

  lowerCamelCase			: Union[str, Any]   =    {left_most_point, right_most_point}
  lowerCamelCase			: int   =    []
  lowerCamelCase			: List[Any]   =    []

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

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

  _construct_hull(SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_     )
  _construct_hull(SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_     )

  return sorted(SCREAMING_SNAKE_CASE_     )






def 		lowercase_(  SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_     ):


  '''simple docstring'''

  if points:
    lowerCamelCase			: List[str]   =    None
    lowerCamelCase			: List[Any]   =    float("-inf"     )
    lowerCamelCase			: Optional[int]   =    []

    for p in points:
      lowerCamelCase			: Optional[Any]   =    _det(SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_     )

      if det > 0:
        candidate_points.append(SCREAMING_SNAKE_CASE_     )

        if det > extreme_point_distance:
          lowerCamelCase			: List[Any]   =    det
          lowerCamelCase			: Optional[int]   =    p

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






def 		lowercase_(  SCREAMING_SNAKE_CASE_     ):


  '''simple docstring'''

  lowerCamelCase			: Optional[int]   =    sorted(_validate_input(SCREAMING_SNAKE_CASE_     )     )
  lowerCamelCase			: Dict   =    len(SCREAMING_SNAKE_CASE_     )

  lowerCamelCase			: Optional[int]   =    points[:2]
  for i in range(2				,						SCREAMING_SNAKE_CASE_     ):
    lowerCamelCase			: List[Any]   =    _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			: Optional[int]   =    points[i]
  i += 1

  for j in range(SCREAMING_SNAKE_CASE_				,						SCREAMING_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(SCREAMING_SNAKE_CASE_     ) > 3 else convex_hull     )






def 		lowercase_(  ):


  '''simple docstring'''

  lowerCamelCase			: 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)]
  lowerCamelCase			: List[str]   =    convex_hull_bf(SCREAMING_SNAKE_CASE_     )

  lowerCamelCase			: Any   =    convex_hull_recursive(SCREAMING_SNAKE_CASE_     )
  assert results_bf == results_recursive

  lowerCamelCase			: Dict   =    convex_hull_melkman(SCREAMING_SNAKE_CASE_     )
  assert results_bf == results_melkman

  print(SCREAMING_SNAKE_CASE_     )


if __name__ == "__main__":
   main()







def 		lowercase_(  SCREAMING_SNAKE_CASE_				,						SCREAMING_SNAKE_CASE_     ):


  '''simple docstring'''

  sequence.append(SCREAMING_SNAKE_CASE_     )

  lowerCamelCase			: str   =    0
  while sequence[index] != target:
    index += 1

  sequence.pop()

  if index == len(SCREAMING_SNAKE_CASE_     ):
    return None

  return index


if __name__ == "__main__":
   _snake_case					=					input('''Enter numbers separated by comma:\n''').strip()
   _snake_case					=					[int(item) for item in user_input.split(''',''')]

   _snake_case					=					input('''Enter a single number to be found in the list:\n''')
   _snake_case					=					int(target_input)
   _snake_case					=					sentinel_linear_search(sequence, target)
   if result is not None:
      print(f'''{target} found at positions: {result}''')
   else:
      print('''Not found''')



