


'''simple docstring'''



from __future__ import annotations

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



class       a							(						_a			):


     """simple docstring"""






     SCREAMING_SNAKE_CASE			:					int				  =  1
     SCREAMING_SNAKE_CASE			:					Optional[int]				  =  2
     SCREAMING_SNAKE_CASE			:					Tuple				  =  3

     def __repr__(							self	:						Any     )		->				Optional[int]:
          return f'{self.__class__.__name__}.{self.name}'




def       _a      (		_lowercase   :      tuple[int, int]     ,   _lowercase   :      int     ,   _lowercase   :      int    ):





     '''simple docstring'''






     __UpperCAmelCase				,  __UpperCAmelCase      :					Dict    				=     point
     return degrees(atana(y - miny     ,   x - minx    )    )




def       _a      (		_lowercase   :      tuple[int, int]     ,   _lowercase   :      tuple[int, int]     ,   _lowercase   :      tuple[int, int]    ):





     '''simple docstring'''






     __UpperCAmelCase				,  __UpperCAmelCase      :					Union[str, Any]    				=     starting
     __UpperCAmelCase				,  __UpperCAmelCase      :					List[Any]    				=     via
     __UpperCAmelCase				,  __UpperCAmelCase      :					Optional[int]    				=     target
     __UpperCAmelCase      :					List[Any]    				=     degrees(atana(ya - ya     ,   xa - xa    )    )
     via_angle %= 360
     __UpperCAmelCase      :					Tuple    				=     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       _a      (		_lowercase   :      list[tuple[int, int]]    ):





     '''simple docstring'''






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

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

     __UpperCAmelCase      :					List[str]    				=     sorted(_lowercase     ,   key=lambda _lowercase    : angle_comparer(_lowercase     ,   _lowercase     ,   _lowercase    )    )
     # 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      :					Optional[int]    				=     Direction.left

     for i in range(3     ,   len(_lowercase    )    ):
          while True:
               __UpperCAmelCase      :					Any    				=     stack[-2]
               __UpperCAmelCase      :					int    				=     stack[-1]
               __UpperCAmelCase      :					str    				=     sorted_points[i]
               __UpperCAmelCase      :					Optional[int]    				=     check_direction(_lowercase     ,   _lowercase     ,   _lowercase    )

               if next_direction == Direction.left:
                    __UpperCAmelCase      :					Tuple    				=     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(_lowercase    )



'''simple docstring'''



import math
from decimal import Decimal, getcontext




def       _a      (		_lowercase   :      int = 4000000    ):





     '''simple docstring'''






     try:
          __UpperCAmelCase      :					List[str]    				=     int(_lowercase    )
     except (TypeError, ValueError):
          raise TypeError('''Parameter n must be int or castable to int.'''    )
     if n <= 0:
          raise ValueError('''Parameter n must be greater than or equal to one.'''    )
     __UpperCAmelCase      :					Tuple    				=     100
     __UpperCAmelCase      :					Optional[Any]    				=     (Decimal(5    ) ** Decimal(0.5    ) + 1) / Decimal(2    )

     __UpperCAmelCase      :					Tuple    				=     (math.floor(math.log(n * (phi + 2)     ,   _lowercase    ) - 1    ) // 3) * 3 + 2
     __UpperCAmelCase      :					List[Any]    				=     Decimal(round(phi ** Decimal(index + 1    )    )    ) / (phi + 2)
     __UpperCAmelCase      :					List[str]    				=     num // 2
     return int(_lowercase    )


if __name__ == "__main__":
      print(f"""{solution() = }""")