
import math




def 		A						(  SCREAMING_SNAKE_CASE							):




			"""simple docstring"""


			if number == 0:
						UpperCAmelCase__		:List[Any] 		=    'Zero'
						return words
			else:
						UpperCAmelCase__		:List[Any] 		=    math.logaa(__lowerCamelCase							)
						UpperCAmelCase__		:Tuple 		=    digits + 1
						UpperCAmelCase__		:int 		=    {}
						UpperCAmelCase__		:Tuple 		=    ''
						UpperCAmelCase__		:int 		=    'One'
						UpperCAmelCase__		:Dict 		=    'Two'
						UpperCAmelCase__		:List[Any] 		=    'Three'
						UpperCAmelCase__		:Optional[Any] 		=    'Four'
						UpperCAmelCase__		:str 		=    'Five'
						UpperCAmelCase__		:Tuple 		=    'Six'
						UpperCAmelCase__		:Dict 		=    'Seven'
						UpperCAmelCase__		:int 		=    'Eight'
						UpperCAmelCase__		:Tuple 		=    'Nine'

						UpperCAmelCase__		:Optional[Any] 		=    {}
						UpperCAmelCase__		:Tuple 		=    ''
						UpperCAmelCase__		:List[str] 		=    'Twenty'
						UpperCAmelCase__		:Optional[int] 		=    'Thirty'
						UpperCAmelCase__		:int 		=    'Forty'
						UpperCAmelCase__		:str 		=    'Fifty'
						UpperCAmelCase__		:List[Any] 		=    'Sixty'
						UpperCAmelCase__		:str 		=    'Seventy'
						UpperCAmelCase__		:Union[str, Any] 		=    'Eighty'
						UpperCAmelCase__		:int 		=    'Ninety'

						UpperCAmelCase__		:Any 		=    {}
						UpperCAmelCase__		:Any 		=    'Ten'
						UpperCAmelCase__		:Optional[int] 		=    'Eleven'
						UpperCAmelCase__		:List[Any] 		=    'Twelve'
						UpperCAmelCase__		:Tuple 		=    'Thirteen'
						UpperCAmelCase__		:str 		=    'Fourteen'
						UpperCAmelCase__		:Tuple 		=    'Fifteen'
						UpperCAmelCase__		:List[str] 		=    'Sixteen'
						UpperCAmelCase__		:Optional[Any] 		=    'Seventeen'
						UpperCAmelCase__		:Tuple 		=    'Eighteen'
						UpperCAmelCase__		:Any 		=    'Nineteen'

						UpperCAmelCase__		:Union[str, Any] 		=    {}
						UpperCAmelCase__		:int 		=    'Hundred,'
						UpperCAmelCase__		:str 		=    'Thousand,'
						UpperCAmelCase__		:Tuple 		=    'Lakh,'
						UpperCAmelCase__		:Optional[Any] 		=    'Crore,'

						UpperCAmelCase__		:Any 		=    number
						UpperCAmelCase__		:Union[str, Any] 		=    ''
						UpperCAmelCase__		:int 		=    0
						UpperCAmelCase__		:Any 		=    int(__lowerCamelCase							)
						while counter < digits:
									UpperCAmelCase__		:List[Any] 		=    temp_num % 10
									if counter % 2 == 0:
												UpperCAmelCase__		:Optional[Any] 		=    ''
												if counter in placevalue and current != 0:
															UpperCAmelCase__		:Optional[int] 		=    placevalue[counter]
												if counter == 2:
															UpperCAmelCase__		:Tuple 		=    singles[current] + addition + words
												elif counter == 0:
															if ((temp_num % 100) // 10) == 1:
																		UpperCAmelCase__		:Union[str, Any] 		=    teens[current] + addition + words
																		UpperCAmelCase__		:Any 		=    temp_num // 10
																		counter += 1
															else:
																		UpperCAmelCase__		:Optional[int] 		=    singles[current] + addition + words

												else:
															UpperCAmelCase__		:int 		=    doubles[current] + addition + words

									else:
												if counter == 1:
															if current == 1:
																		UpperCAmelCase__		:Union[str, Any] 		=    teens[number % 10] + words
															else:
																		UpperCAmelCase__		:str 		=    ''
																		if counter in placevalue:
																					UpperCAmelCase__		:str 		=    placevalue[counter]
																		UpperCAmelCase__		:Tuple 		=    doubles[current] + addition + words
												else:
															UpperCAmelCase__		:Union[str, Any] 		=    ''
															if counter in placevalue:
																		if current == 0 and ((temp_num % 100) // 10) == 0:
																					UpperCAmelCase__		:Union[str, Any] 		=    ''
																		else:
																					UpperCAmelCase__		:List[Any] 		=    placevalue[counter]
															if ((temp_num % 100) // 10) == 1:
																		UpperCAmelCase__		:int 		=    teens[current] + addition + words
																		UpperCAmelCase__		:int 		=    temp_num // 10
																		counter += 1
															else:
																		UpperCAmelCase__		:Tuple 		=    singles[current] + addition + words
									counter += 1
									UpperCAmelCase__		:str 		=    temp_num // 10
			return words


if __name__ == "__main__":
						import doctest

						doctest.testmod()







from __future__ import annotations

from collections.abc import Iterable





class 				UpperCamelCase__		:






       """simple docstring"""


       def __init__(							self ,     _A ,     _A	)				->				Union[str, Any]:
          SCREAMING_SNAKE_CASE_	, SCREAMING_SNAKE_CASE_	=      float(_A	), float(_A	)



       def __eq__(							self ,     _A	)				->				str:
          return self.x == other.x and self.y == other.y



       def __ne__(							self ,     _A	)				->				Any:
          return not self == other



       def __gt__(							self ,     _A	)				->				List[Any]:
          if self.x > other.x:
             return True
          elif self.x == other.x:
             return self.y > other.y
          return False



       def __lt__(							self ,     _A	)				->				Optional[int]:
          return not self > other



       def __ge__(							self ,     _A	)				->				int:
          if self.x > other.x:
             return True
          elif self.x == other.x:
             return self.y >= other.y
          return False



       def __le__(							self ,     _A	)				->				str:
          if self.x < other.x:
             return True
          elif self.x == other.x:
             return self.y <= other.y
          return False



       def __repr__(							self	)				->				str:
          return F'''({self.x}, {self.y})'''




       def __hash__(							self	)				->				List[str]:
          return hash(self.x	)







def  A__ (       __lowerCamelCase,  ):
   SCREAMING_SNAKE_CASE_	=      []
   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  A__ (       __lowerCamelCase						):

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

   if not points:
      SCREAMING_SNAKE_CASE_	=      F'''Expecting a list of points but got {points}'''
      raise ValueError(__lowerCamelCase						)

   return _construct_points(__lowerCamelCase						)


def  A__ (       __lowerCamelCase,  __lowerCamelCase,  __lowerCamelCase						):
   SCREAMING_SNAKE_CASE_	=      (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__ (       __lowerCamelCase						):
   SCREAMING_SNAKE_CASE_	=      sorted(_validate_input(__lowerCamelCase						)						)
   SCREAMING_SNAKE_CASE_	=      len(__lowerCamelCase						)
   SCREAMING_SNAKE_CASE_	=      set()

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

               if det_k > 0:
                  SCREAMING_SNAKE_CASE_	=      True
               elif det_k < 0:
                  SCREAMING_SNAKE_CASE_	=      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_	=      False
                     break

            if points_left_of_ij and points_right_of_ij:
               SCREAMING_SNAKE_CASE_	=      False
               break

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

   return sorted(__lowerCamelCase						)


def  A__ (       __lowerCamelCase						):
   SCREAMING_SNAKE_CASE_	=      sorted(_validate_input(__lowerCamelCase						)						)
   SCREAMING_SNAKE_CASE_	=      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

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

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

   for i in range(1,  n - 1						):
      SCREAMING_SNAKE_CASE_	=      _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  A__ (       __lowerCamelCase,  __lowerCamelCase,  __lowerCamelCase,  __lowerCamelCase						):
   if points:
      SCREAMING_SNAKE_CASE_	=      None
      SCREAMING_SNAKE_CASE_	=      float('''-inf'''						)
      SCREAMING_SNAKE_CASE_	=      []

      for p in points:
         SCREAMING_SNAKE_CASE_	=      _det(__lowerCamelCase,  __lowerCamelCase,  __lowerCamelCase						)

         if det > 0:
            candidate_points.append(__lowerCamelCase						)

            if det > extreme_point_distance:
               SCREAMING_SNAKE_CASE_	=      det
               SCREAMING_SNAKE_CASE_	=      p

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


def  A__ (       __lowerCamelCase						):
   SCREAMING_SNAKE_CASE_	=      sorted(_validate_input(__lowerCamelCase						)						)
   SCREAMING_SNAKE_CASE_	=      len(__lowerCamelCase						)

   SCREAMING_SNAKE_CASE_	=      points[:2]
   for i in range(2,  __lowerCamelCase						):
      SCREAMING_SNAKE_CASE_	=      _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_	=      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  A__ (       ):
   SCREAMING_SNAKE_CASE_	=      [
       (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_	=      convex_hull_bf(__lowerCamelCase						)

   SCREAMING_SNAKE_CASE_	=      convex_hull_recursive(__lowerCamelCase						)
   assert results_bf == results_recursive

   SCREAMING_SNAKE_CASE_	=      convex_hull_melkman(__lowerCamelCase						)
   assert results_bf == results_melkman

   print(__lowerCamelCase						)


if __name__ == "__main__":
   main()


