'''simple docstring'''


from __future__ import annotations

from collections import deque
from enum import Enum
from math import atana, degrees
from sys import maxsize






class 	__SCREAMING_SNAKE_CASE		(			lowerCamelCase_			):







  '''simple docstring'''






  lowerCamelCase_						:Tuple		       =						1
  lowerCamelCase_						:List[Any]		       =						2
  lowerCamelCase_						:Any		       =						3


  def __repr__(					self		):




       '''simple docstring'''

       return F'''{self.__class__.__name__}.{self.name}'''







def    _lowerCamelCase       ( lowerCamelCase_   :				tuple[int, int]      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int  ):



     """simple docstring"""





     UpperCAmelCase_  ,     UpperCAmelCase_				:  Tuple  = point
     return degrees(atana(y - miny      ,	x - minx  )  )


def    _lowerCamelCase       ( lowerCamelCase_   :				tuple[int, int]      ,	lowerCamelCase_   :				tuple[int, int]      ,	lowerCamelCase_   :				tuple[int, int]  ):



     """simple docstring"""





     UpperCAmelCase_  ,     UpperCAmelCase_				:  Union[str, Any]  = starting
     UpperCAmelCase_  ,     UpperCAmelCase_				:  Optional[Any]  = via
     UpperCAmelCase_  ,     UpperCAmelCase_				:  List[Any]  = target
     UpperCAmelCase_				:  Any  = degrees(atana(ya - ya      ,	xa - xa  )  )
     via_angle %= 360
     UpperCAmelCase_				:  Union[str, Any]  = degrees(atana(ya - ya      ,	xa - xa  )  )
     target_angle %= 360
     # t-
     #  \ \
     #   \ v
     #    \|
     #     s
     # via_angle is always lower than target_angle, if direction is left.
     # If they are same, it means they are on a same line of convex hull.
     if target_angle > via_angle:
          return Direction.left
     elif target_angle == via_angle:
          return Direction.straight
     else:
          return Direction.right





def    _lowerCamelCase       ( lowerCamelCase_   :				list[tuple[int, int]]  ):



     """simple docstring"""





     if len(lowerCamelCase_  ) <= 2:
          # There is no convex hull
          raise ValueError('graham_scan: argument must contain more than 3 points.'  )
     if len(lowerCamelCase_  ) == 3:
          return points
     # find the lowest and the most left point
     UpperCAmelCase_				:  Union[str, Any]  = 0
     UpperCAmelCase_  ,     UpperCAmelCase_				:  Any  = maxsize, maxsize
     for i, point in enumerate(lowerCamelCase_  ):
          UpperCAmelCase_				:  Optional[Any]  = point[0]
          UpperCAmelCase_				:  Dict  = point[1]
          if y < miny:
               UpperCAmelCase_				:  List[Any]  = y
               UpperCAmelCase_				:  Union[str, Any]  = x
               UpperCAmelCase_				:  Union[str, Any]  = i
          if y == miny and x < minx:
               UpperCAmelCase_				:  Dict  = x
               UpperCAmelCase_				:  Optional[Any]  = i

    # remove the lowest and the most left point from points for preparing for sort
     points.pop(lowerCamelCase_  )

     UpperCAmelCase_				:  List[str]  = sorted(lowerCamelCase_      ,	key=lambda lowerCamelCase_  : angle_comparer(lowerCamelCase_      ,	lowerCamelCase_      ,	lowerCamelCase_  )  )
     # This insert actually costs complexity,
     # and you should instead add (minx, miny) into stack later.
     # I'm using insert just for easy understanding.
     sorted_points.insert(0      ,	(minx, miny)  )

     UpperCAmelCase_				:  deque[tuple[int, int]]  = deque()
     stack.append(sorted_points[0]  )
     stack.append(sorted_points[1]  )
     stack.append(sorted_points[2]  )
     # In any ways, the first 3 points line are towards left.
     # Because we sort them the angle from minx, miny.
     UpperCAmelCase_				:  List[Any]  = Direction.left

     for i in range(3      ,	len(lowerCamelCase_  )  ):
          while True:
               UpperCAmelCase_				:  Union[str, Any]  = stack[-2]
               UpperCAmelCase_				:  Any  = stack[-1]
               UpperCAmelCase_				:  Optional[Any]  = sorted_points[i]
               UpperCAmelCase_				:  List[str]  = check_direction(lowerCamelCase_      ,	lowerCamelCase_      ,	lowerCamelCase_  )

               if next_direction == Direction.left:
                    UpperCAmelCase_				:  List[str]  = Direction.left
                    break
               if next_direction == Direction.straight:
                    if current_direction == Direction.left:
                         # We keep current_direction as left.
                         # Because if the straight line keeps as straight,
                         # we want to know if this straight line is towards left.
                         break
                    elif current_direction == Direction.right:
                         # If the straight line is towards right,
                         # every previous points on those straigh line is not convex hull.
                         stack.pop()
               if next_direction == Direction.right:
                    stack.pop()
          stack.append(sorted_points[i]  )
     return list(lowerCamelCase_  )











'''simple docstring'''
from __future__ import annotations


def 	lowercase_						( lowerCAmelCase__						:	int      ,				lowerCAmelCase__						:	int      ,				lowerCAmelCase__						:	int			):

			"""simple docstring"""





			assert (
			    c % greatest_common_divisor(lowerCAmelCase__      ,				lowerCAmelCase__			) == 0
			)  # greatest_common_divisor(a,b) function implemented below
			((__UpperCAmelCase)					,	(__UpperCAmelCase)					,	(__UpperCAmelCase)) :   int    					=  extended_gcd(lowerCAmelCase__      ,				lowerCAmelCase__			)  # extended_gcd(a,b) function implemented below
			__UpperCAmelCase :   Union[str, Any]    					=  c / d
			return (r * x, r * y)


def 	lowercase_						( lowerCAmelCase__						:	int      ,				lowerCAmelCase__						:	int      ,				lowerCAmelCase__						:	int      ,				lowerCAmelCase__						:	int = 2			):

			"""simple docstring"""





			((__UpperCAmelCase)					,	(__UpperCAmelCase)) :   List[str]    					=  diophantine(lowerCAmelCase__      ,				lowerCAmelCase__      ,				lowerCAmelCase__			)  # Initial value
			__UpperCAmelCase :   List[str]    					=  greatest_common_divisor(lowerCAmelCase__      ,				lowerCAmelCase__			)
			__UpperCAmelCase :   List[str]    					=  a // d
			__UpperCAmelCase :   Tuple    					=  b // d

			for i in range(lowerCAmelCase__			):
						__UpperCAmelCase :   List[Any]    					=  xa + i * q
						__UpperCAmelCase :   Tuple    					=  ya - i * p
						print(lowerCAmelCase__      ,				lowerCAmelCase__			)


def 	lowercase_						( lowerCAmelCase__						:	int      ,				lowerCAmelCase__						:	int			):

			"""simple docstring"""





			if a < b:
						__UpperCAmelCase					,	__UpperCAmelCase :   List[str]    					=  b, a

			while a % b != 0:
						__UpperCAmelCase					,	__UpperCAmelCase :   Optional[int]    					=  b, a % b

			return b


def 	lowercase_						( lowerCAmelCase__						:	int      ,				lowerCAmelCase__						:	int			):

			"""simple docstring"""





			assert a >= 0 and b >= 0

			if b == 0:
						__UpperCAmelCase					,	__UpperCAmelCase					,	__UpperCAmelCase :   List[str]    					=  a, 1, 0
			else:
						((__UpperCAmelCase)					,	(__UpperCAmelCase)					,	(__UpperCAmelCase)) :   str    					=  extended_gcd(lowerCAmelCase__      ,				a % b			)
						__UpperCAmelCase :   Optional[Any]    					=  q
						__UpperCAmelCase :   Dict    					=  p - q * (a // b)

			assert a % d == 0 and b % d == 0
			assert d == a * x + b * y

			return (d, x, y)


if __name__ == "__main__":
					from doctest import testmod

					testmod(name='''diophantine''', verbose=True)
					testmod(name='''diophantine_all_soln''', verbose=True)
					testmod(name='''extended_gcd''', verbose=True)
					testmod(name='''greatest_common_divisor''', verbose=True)




