from __future__ import annotations

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

class 				_lowercase						(    UpperCAmelCase__ ):






    '''simple docstring'''
    SCREAMING_SNAKE_CASE:							Optional[Any]        =						1
    SCREAMING_SNAKE_CASE:							Any        =						2
    SCREAMING_SNAKE_CASE:							str        =						3




    def __repr__(						self   ):
       return F'''{self.__class__.__name__}.{self.name}'''





def 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase				):
   lowerCAmelCase_:       str         =  point
   return degrees(atana(y - miny       ,							x - minx				)				)
def 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase				):
   lowerCAmelCase_:       List[str]         =  starting
   lowerCAmelCase_:       str         =  via
   lowerCAmelCase_:       Optional[Any]         =  target
   lowerCAmelCase_:       Optional[Any]         =  degrees(atana(ya - ya       ,							xa - xa				)				)
   via_angle %= 360
   lowerCAmelCase_:       str         =  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 		snake_case__  (		lowercase				):

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

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

   lowerCAmelCase_:       str         =  sorted(UpperCamelCase__       ,							key=lambda lowercase				: angle_comparer(UpperCamelCase__       ,							UpperCamelCase__       ,							UpperCamelCase__				)				)
   # 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)				)

   lowerCAmelCase_:       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.
   lowerCAmelCase_:       List[Any]         =  Direction.left

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

         if next_direction == Direction.left:
            lowerCAmelCase_:       Union[str, Any]         =  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(UpperCamelCase__				)
from .balanced_parentheses import balanced_parentheses
from .stack import Stack
def 		snake_case__  (		lowercase				):
   return {"+": 1, "-": 1, "*": 2, "/": 2, "^": 3}.get(lowercase       ,							-1				)
def 		snake_case__  (		lowercase				):
   if not balanced_parentheses(lowercase				):
      raise ValueError("Mismatched parentheses"				)
   lowerCAmelCase_:       Stack[str]         =  Stack()
   lowerCAmelCase_:       Union[str, Any]         =  []
   for char in expression_str:
      if char.isalpha() or char.isdigit():
         postfix.append(lowercase				)
      elif char == "(":
         stack.push(lowercase				)
      elif char == ")":
         while not stack.is_empty() and stack.peek() != "(":
            postfix.append(stack.pop()				)
         stack.pop()
      else:
         while not stack.is_empty() and precedence(lowercase				) <= precedence(stack.peek()				):
            postfix.append(stack.pop()				)
         stack.push(lowercase				)
   while not stack.is_empty():
      postfix.append(stack.pop()				)
   return " ".join(lowercase				)


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

 testmod()
 a     :			int					    =						"""a+b*(c^d-e)^(f+g*h)-i"""

 print("""Infix to Postfix Notation demonstration:\n""")
 print("""Infix notation: """ + expression)
 print("""Postfix notation: """ + infix_to_postfix(expression))