


'''simple docstring'''

import math







class 			UpperCAmelCase			:

 def __init__(			self			:						List[Any],       a_			:						int,       a_			:						int,       a_			:						int,       a_			:						Tuple       ):





    """simple docstring"""



    UpperCamelCase__			    =    x
    UpperCamelCase__			    =    y
    UpperCamelCase__			    =    step_size
    UpperCamelCase__			    =    function_to_optimize

 def 		lowercase_    (			self			:						Dict       ):





    """simple docstring"""



    return self.function(self.x,       self.y       )

 def 		lowercase_    (			self			:						Union[str, Any]       ):





    """simple docstring"""



    UpperCamelCase__			    =    self.step_size
    return [
        SearchProblem(a_,       a_,       a_,       self.function       )
        for x, y in (
            (self.x - step_size, self.y - step_size),
            (self.x - step_size, self.y),
            (self.x - step_size, self.y + step_size),
            (self.x, self.y - step_size),
            (self.x, self.y + step_size),
            (self.x + step_size, self.y - step_size),
            (self.x + step_size, self.y),
            (self.x + step_size, self.y + step_size),
        )
    ]

 def __hash__(			self			:						Optional[Any]       ):





    """simple docstring"""



    return hash(str(self       )       )

 def __eq__(			self			:						int,       a_			:						Tuple       ):





    """simple docstring"""



    if isinstance(a_,       a_       ):
       return hash(str(self       )       ) == hash(str(a_       )       )
    return False

 def __str__(			self			:						List[str]       ):





    """simple docstring"""



    return f'x: {self.x} y: {self.y}'







def 							SCREAMING_SNAKE_CASE__(		_UpperCamelCase   :					Tuple     ,		_UpperCamelCase   :					bool = True     ,		_UpperCamelCase   :					float = math.inf     ,		_UpperCamelCase   :					float = -math.inf     ,		_UpperCamelCase   :					float = math.inf     ,		_UpperCamelCase   :					float = -math.inf     ,		_UpperCamelCase   :					bool = False     ,		_UpperCamelCase   :					int = 1_00_00     ,		)		->       SearchProblem:





   '''simple docstring'''







   UpperCamelCase__			    =    search_prob
   UpperCamelCase__			    =    []  # list to store the current score at each iteration
   UpperCamelCase__			    =    0
   UpperCamelCase__			    =    False
   UpperCamelCase__			    =    set()
   while not solution_found and iterations < max_iter:
      visited.add(_UpperCamelCase )
      iterations += 1
      UpperCamelCase__			    =    current_state.score()
      scores.append(_UpperCamelCase )
      UpperCamelCase__			    =    current_state.get_neighbors()
      UpperCamelCase__			    =    -math.inf
      UpperCamelCase__			    =    math.inf
      UpperCamelCase__			    =    None  # to hold the next best neighbor
      for neighbor in neighbors:
         if neighbor in visited:
            continue  # do not want to visit the same state again
         if (
             neighbor.x > max_x
             or neighbor.x < min_x
             or neighbor.y > max_y
             or neighbor.y < min_y
         ):
            continue  # neighbor outside our bounds
         UpperCamelCase__			    =    neighbor.score() - current_score
         if find_max:  # finding max
            # going to direction with greatest ascent
            if change > max_change and change > 0:
               UpperCamelCase__			    =    change
               UpperCamelCase__			    =    neighbor
         else:  # finding min
            # to direction with greatest descent
            if change < min_change and change < 0:
               UpperCamelCase__			    =    change
               UpperCamelCase__			    =    neighbor
      if next_state is not None:
         # we found at least one neighbor which improved the current state
         UpperCamelCase__			    =    next_state
      else:
         # since we have no neighbor that improves the solution we stop the search
         UpperCamelCase__			    =    True

   if visualization:
      from matplotlib import pyplot as plt

      plt.plot(range(_UpperCamelCase )     ,		_UpperCamelCase )
      plt.xlabel("Iterations" )
      plt.ylabel("Function values" )
      plt.show()

   return current_state


if __name__ == "__main__":
       import doctest

       doctest.testmod()



       def 							SCREAMING_SNAKE_CASE__(		_UpperCamelCase   :					Tuple     ,		_UpperCamelCase   :					Any )		->       Any:





          '''simple docstring'''







          return (x**2) + (y**2)

       # starting the problem with initial coordinates (3, 4)
       __lowercase:		Tuple    						=  SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_fa)
       __lowercase:		Optional[Any]    						=  hill_climbing(prob, find_max=False)
       print(
           "The minimum score for f(x, y) = x^2 + y^2 found via hill climbing: "
           F"""{local_min.score()}"""
       )

       # starting the problem with initial coordinates (12, 47)
       __lowercase:		List[str]    						=  SearchProblem(x=12, y=47, step_size=1, function_to_optimize=test_fa)
       __lowercase:		str    						=  hill_climbing(
           prob, find_max=False, max_x=100, min_x=5, max_y=50, min_y=-5, visualization=True
       )
       print(
           "The minimum score for f(x, y) = x^2 + y^2 with the domain 100 > x > 5 "
           F"""and 50 > y > - 5 found via hill climbing: {local_min.score()}"""
       )



       def 							SCREAMING_SNAKE_CASE__(		_UpperCamelCase   :					int     ,		_UpperCamelCase   :					Union[str, Any] )		->       Any:





          '''simple docstring'''







          return (3 * x**2) - (6 * y)

       __lowercase:		Dict    						=  SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_fa)
       __lowercase:		Union[str, Any]    						=  hill_climbing(prob, find_max=True)






       print(
           "The maximum score for f(x, y) = x^2 + y^2 found via hill climbing: "
           F"""{local_min.score()}"""
       )

'''simple docstring'''
import itertools

def     UpperCamelCase_  (						__SCREAMING_SNAKE_CASE = 1_0_0_0_0_0_0							):
    """simple docstring"""






    snake_case_			:     Optional[int]        =    [1]

    for i in itertools.count(len(__SCREAMING_SNAKE_CASE							)							):
        snake_case_			:     str        =    0
        for j in itertools.count(1							):
            snake_case_			:     List[str]        =    -1 if j % 2 == 0 else +1
            snake_case_			:     str        =    (j * j * 3 - j) // 2
            if index > i:
                break
            item += partitions[i - index] * sign
            item %= number
            index += j
            if index > i:
                break
            item += partitions[i - index] * sign
            item %= number

        if item == 0:
            return i
        partitions.append(__SCREAMING_SNAKE_CASE							)

    return 0


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  print(f"""{solution() = }""")




