code: 
  '''simple docstring'''
  
  from __future__ import annotations
  
  from abc import abstractmethod
  from collections.abc import Iterable
  from typing import Generic, Protocol, TypeVar
  
  
  class   a_  (_a    ):
  
  
  
  
         @abstractmethod
         def __lt__(     self   ,   snake_case_  ):
           pass
  
  
  
  
         @abstractmethod
         def __gt__(     self   ,   snake_case_  ):
           pass
  
  
  
  
  
  
  
         @abstractmethod
         def __eq__(     self   ,   snake_case_  ):
           pass
  
  
  
  UpperCamelCase_     			= TypeVar("""T""", bound=Comparable)
  
  
  class   a_  (Generic[T]    ):
  
  
  
  
         def __init__(     self  ):
           _lowerCAmelCase       :  list[T]   				=					[]
           _lowerCAmelCase       :  int   				=					0
  
  
  
  
         def __repr__(     self  ):
           return str(self.h  )
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_  ):
           if child_idx > 0:
             return (child_idx - 1) // 2
           return None
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_  ):
           _lowerCAmelCase       :  Dict   				=					2 * parent_idx + 1
           if left_child_index < self.heap_size:
             return left_child_index
           return None
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_  ):
           _lowerCAmelCase       :  Union[str, Any]   				=					2 * parent_idx + 2
           if right_child_index < self.heap_size:
             return right_child_index
           return None
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_  ):
           if index < self.heap_size:
             _lowerCAmelCase       :  int   				=					index
             _lowerCAmelCase       :  Tuple   				=					self.left_child_idx(snake_case_  )
             _lowerCAmelCase       :  Tuple   				=					self.right_child_idx(snake_case_  )
             # check which child is larger than its parent
             if left_child is not None and self.h[left_child] > self.h[violation]:
               _lowerCAmelCase       :  Tuple   				=					left_child
             if right_child is not None and self.h[right_child] > self.h[violation]:
               _lowerCAmelCase       :  Any   				=					right_child
             # if violation indeed exists
             if violation != index:
               # swap to fix the violation
               _lowerCAmelCase       :  List[str]   				=					self.h[index], self.h[violation]
               # fix the subsequent violation recursively if any
               self.max_heapify(snake_case_  )
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_  ):
           _lowerCAmelCase       :  Tuple   				=					list(snake_case_  )
           _lowerCAmelCase       :  Union[str, Any]   				=					len(self.h  )
           if self.heap_size > 1:
             # max_heapify from right to left but exclude leaves (last level)
             for i in range(self.heap_size // 2 - 1   ,   -1   ,   -1  ):
               self.max_heapify(snake_case_  )
  
  
  
  
         def 							__UpperCamelCase       (     self  ):
           if self.heap_size >= 2:
             _lowerCAmelCase       :  List[str]   				=					self.h[0]
             _lowerCAmelCase       :  str   				=					self.h.pop(-1  )
             self.heap_size -= 1
             self.max_heapify(0  )
             return me
           elif self.heap_size == 1:
             self.heap_size -= 1
             return self.h.pop(-1  )
           else:
             raise Exception("""Empty heap"""  )
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_  ):
           self.h.append(snake_case_  )
           _lowerCAmelCase       :  Union[str, Any]   				=					(self.heap_size - 1) // 2
           self.heap_size += 1
           while idx >= 0:
             self.max_heapify(snake_case_  )
             _lowerCAmelCase       :  Optional[int]   				=					(idx - 1) // 2
  
  
  
  
  
  
  
         def 							__UpperCamelCase       (     self  ):
           _lowerCAmelCase       :  Tuple   				=					self.heap_size
           for j in range(size - 1   ,   0   ,   -1  ):
             _lowerCAmelCase       :  int   				=					self.h[j], self.h[0]
             self.heap_size -= 1
             self.max_heapify(0  )
           _lowerCAmelCase       :  Union[str, Any]   				=					size
  
  
  if __name__ == "__main__":
       import doctest
  
       # run doc test
       doctest.testmod()
  
       # demo
       for unsorted in [
           [0],
           [2],
           [3, 5],
           [5, 3],
           [5, 5],
           [0, 0, 0, 0],
           [1, 1, 1, 1],
           [2, 2, 3, 5],
           [0, 2, 2, 3, 5],
           [2, 5, 3, 0, 2, 3, 0, 3],
           [6, 1, 2, 7, 9, 3, 4, 5, 10, 8],
           [1_03, 9, 1, 7, 11, 15, 25, 2_01, 2_09, 1_07, 5],
           [-45, -2, -5],
       ]:
            print(F'unsorted array: {unsorted}')
  
            UpperCamelCase_     			= Heap()
            heap.build_max_heap(unsorted)
            print(F'after build heap: {heap}')
  
            print(F'max value: {heap.extract_max()}')
            print(F'after max value removed: {heap}')
  
            heap.insert(1_00)
            print(F'after new value 100 inserted: {heap}')
  
            heap.heap_sort()
            print(F'heap-sorted array: {heap}\n')
  
  
  
style_context: 
  '''simple docstring'''
  
  import random
  
  
  class   a_  :
  
  
  
  
         def __init__(     self   ,   snake_case_   ,   snake_case_  ):
           _lowerCAmelCase       :  List[str]   				=					x
           _lowerCAmelCase       :  List[str]   				=					y
  
  
  
  
         def 							__UpperCamelCase       (     self  ):
           return (self.x**2 + self.y**2) <= 1
  
  
  
  
  
  
  
         @classmethod
         def 							__UpperCamelCase       (     cls  ):
           return cls(x=random.random()   ,   y=random.random()  )
  
  
  
  def 	_UpperCAmelCase				(  _lowerCamelCase						:       int		)    ->						float:
    if number_of_simulations < 1:
      raise ValueError("""At least one simulation is necessary to estimate PI."""		)
  
    _lowerCAmelCase       :  Tuple   				=					0
    for _ in range(_lowerCamelCase		):
      _lowerCAmelCase       :  Union[str, Any]   				=					Point.random_unit_square()
  
      if random_point.is_in_unit_circle():
        number_in_unit_circle += 1
  
    return 4 * number_in_unit_circle / number_of_simulations
  
  
  if __name__ == "__main__":
       # import doctest
  
       # doctest.testmod()
       from math import pi
  
       UpperCamelCase_     			=       """Please enter the desired number of Monte Carlo simulations: """
       UpperCamelCase_     			=       estimate_pi(int(input(prompt).strip()))
       print(F'An estimate of PI is {my_pi} with an error of {abs(my_pi - pi)}')
  
  
  