
'''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')




'''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)}')


