

'''simple docstring'''

from __future__ import annotations

from collections.abc import Iterable





class 							_snake_case	:
   def __init__(	self     : Optional[Any]       ,SCREAMING_SNAKE_CASE__     : Optional[Any]       ,SCREAMING_SNAKE_CASE__     : Optional[int]							):
     SCREAMING_SNAKE_CASE:Optional[int]   =   float(SCREAMING_SNAKE_CASE__							), float(SCREAMING_SNAKE_CASE__							)
   def __eq__(	self     : Optional[Any]       ,SCREAMING_SNAKE_CASE__     : Optional[int]							):
     return self.x == other.x and self.y == other.y
   def __ne__(	self     : Optional[Any]       ,SCREAMING_SNAKE_CASE__     : Optional[int]							):
     return not self == other
   def __gt__(	self     : str       ,SCREAMING_SNAKE_CASE__     : int							):
     if self.x > other.x:
       return True
     elif self.x == other.x:
       return self.y > other.y
     return False
   def __lt__(	self     : List[Any]       ,SCREAMING_SNAKE_CASE__     : int							):
     return not self > other
   def __ge__(	self     : Any       ,SCREAMING_SNAKE_CASE__     : int							):
     if self.x > other.x:
       return True
     elif self.x == other.x:
       return self.y >= other.y
     return False
   def __le__(	self     : Optional[int]       ,SCREAMING_SNAKE_CASE__     : str							):
     if self.x < other.x:
       return True
     elif self.x == other.x:
       return self.y <= other.y
     return False
   def __repr__(	self     : Union[str, Any]							):
     return F'''({self.x}, {self.y})'''







   def __hash__(	self     : str							):
     return hash(self.x							)





def 				A_					(	snake_case ,    ):
  SCREAMING_SNAKE_CASE: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 				A_					(	snake_case	):

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

  if not points:
    SCREAMING_SNAKE_CASE:Tuple   =   F'''Expecting a list of points but got {points}'''
    raise ValueError(snake_case	)

  return _construct_points(snake_case	)





def 				A_					(	snake_case ,    snake_case ,    snake_case	):
  SCREAMING_SNAKE_CASE:Optional[int]   =   (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 				A_					(	snake_case	):
  SCREAMING_SNAKE_CASE:int   =   sorted(_validate_input(snake_case	)	)
  SCREAMING_SNAKE_CASE:int   =   len(snake_case	)
  SCREAMING_SNAKE_CASE:Union[str, Any]   =   set()

  for i in range(n - 1	):
    for j in range(i + 1 ,    snake_case	):
      SCREAMING_SNAKE_CASE:List[str]   =   False
      SCREAMING_SNAKE_CASE:Optional[Any]   =   True
      for k in range(snake_case	):
        if k != i and k != j:
          SCREAMING_SNAKE_CASE:Union[str, Any]   =   _det(points[i] ,    points[j] ,    points[k]	)

          if det_k > 0:
            SCREAMING_SNAKE_CASE:Union[str, Any]   =   True
          elif det_k < 0:
            SCREAMING_SNAKE_CASE: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]:
              SCREAMING_SNAKE_CASE:Dict   =   False
              break

        if points_left_of_ij and points_right_of_ij:
          SCREAMING_SNAKE_CASE:str   =   False
          break

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

  return sorted(snake_case	)





def 				A_					(	snake_case	):
  SCREAMING_SNAKE_CASE:List[Any]   =   sorted(_validate_input(snake_case	)	)
  SCREAMING_SNAKE_CASE: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

  SCREAMING_SNAKE_CASE:Dict   =   points[0]
  SCREAMING_SNAKE_CASE:int   =   points[n - 1]

  SCREAMING_SNAKE_CASE:Tuple   =   {left_most_point, right_most_point}
  SCREAMING_SNAKE_CASE:Optional[Any]   =   []
  SCREAMING_SNAKE_CASE:Optional[Any]   =   []

  for i in range(1 ,    n - 1	):
    SCREAMING_SNAKE_CASE:Dict   =   _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 				A_					(	snake_case ,    snake_case ,    snake_case ,    snake_case	):
  if points:
    SCREAMING_SNAKE_CASE:int   =   None
    SCREAMING_SNAKE_CASE:Any   =   float("-inf"	)
    SCREAMING_SNAKE_CASE:Union[str, Any]   =   []

    for p in points:
      SCREAMING_SNAKE_CASE:List[str]   =   _det(snake_case ,    snake_case ,    snake_case	)

      if det > 0:
        candidate_points.append(snake_case	)

        if det > extreme_point_distance:
          SCREAMING_SNAKE_CASE:Tuple   =   det
          SCREAMING_SNAKE_CASE:List[str]   =   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 				A_					(	snake_case	):
  SCREAMING_SNAKE_CASE:Union[str, Any]   =   sorted(_validate_input(snake_case	)	)
  SCREAMING_SNAKE_CASE:Optional[Any]   =   len(snake_case	)

  SCREAMING_SNAKE_CASE:Optional[Any]   =   points[:2]
  for i in range(2 ,    snake_case	):
    SCREAMING_SNAKE_CASE:List[str]   =   _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:
      SCREAMING_SNAKE_CASE:Any   =   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 				A_					(	):
  SCREAMING_SNAKE_CASE: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)]
  SCREAMING_SNAKE_CASE:Dict   =   convex_hull_bf(snake_case	)

  SCREAMING_SNAKE_CASE:str   =   convex_hull_recursive(snake_case	)
  assert results_bf == results_recursive

  SCREAMING_SNAKE_CASE:List[Any]   =   convex_hull_melkman(snake_case	)
  assert results_bf == results_melkman

  print(snake_case	)


if __name__ == "__main__":
     main()









'''simple docstring'''

from __future__ import annotations

from collections import Counter





def 				A_					(	snake_case	):
  SCREAMING_SNAKE_CASE:int   =   [True] * n
  SCREAMING_SNAKE_CASE:List[str]   =   False
  SCREAMING_SNAKE_CASE:Tuple   =   False
  SCREAMING_SNAKE_CASE:int   =   True

  for i in range(3 ,    int(n**0.5 + 1	) ,    2	):
    SCREAMING_SNAKE_CASE:str   =   i * 2
    while index < n:
      SCREAMING_SNAKE_CASE:Any   =   False
      SCREAMING_SNAKE_CASE:List[Any]   =   index + i

  SCREAMING_SNAKE_CASE:Union[str, Any]   =   [2]

  for i in range(3 ,    snake_case ,    2	):
    if is_prime[i]:
      primes.append(snake_case	)

  return primes





def 				A_					(	snake_case	):
  SCREAMING_SNAKE_CASE:List[str]   =   str(snake_case	)
  SCREAMING_SNAKE_CASE:Union[str, Any]   =   []
  SCREAMING_SNAKE_CASE:List[str]   =   ["0", "1", "2", "3", "4", "5", "6", "7", "8", "9"]

  for duplicate in Counter(snake_case	) - Counter(set(snake_case	)	):
    SCREAMING_SNAKE_CASE:List[str]   =   [int(number_str.replace(snake_case ,    snake_case	)	) for digit in digits]
    replacements.append(snake_case	)

  return replacements





def 				A_					(	snake_case = 8	):
  SCREAMING_SNAKE_CASE:str   =   set()

  # Filter primes with less than 3 replaceable digits
  SCREAMING_SNAKE_CASE:Tuple   =   {
      x for x in set(prime_sieve(1000000	)	) if len(str(snake_case	)	) - len(set(str(snake_case	)	)	) >= 3
  }

  for prime in primes:
    if prime in numbers_checked:
      continue

    SCREAMING_SNAKE_CASE:List[Any]   =   digit_replacements(snake_case	)

    for family in replacements:
      numbers_checked.update(snake_case	)
      SCREAMING_SNAKE_CASE:Optional[Any]   =   primes.intersection(snake_case	)

      if len(snake_case	) != family_length:
        continue

      return min(snake_case	)

  return -1


if __name__ == "__main__":
    print(solution())






