






import math







class       UpperCAmelCase_						:



      """simple docstring"""
      def __init__(    self   ,    SCREAMING_SNAKE_CASE_			)    ->				int:
          UpperCamelCase	:Optional[Any]     							=       len(_A			)
          UpperCamelCase	:Tuple     							=       [0] * (
              4 * self.N
          )  # approximate the overall size of segment tree with array N
          self.build(1   ,    0   ,    self.N - 1			)
      def    UpperCAmelCase						(    self   ,    SCREAMING_SNAKE_CASE_			)    ->				List[Any]:
          return idx * 2
      def    UpperCAmelCase						(    self   ,    SCREAMING_SNAKE_CASE_			)    ->				Union[str, Any]:
          return idx * 2 + 1
      def    UpperCAmelCase						(    self   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_			)    ->				Optional[Any]:  # noqa: E741
          if l == r:
              UpperCamelCase	:Optional[int]     							=       A[l]
          else:
              UpperCamelCase	:int     							=       (l + r) // 2
              self.build(self.left(_A			)   ,    _A   ,    _A			)
              self.build(self.right(_A			)   ,    mid + 1   ,    _A			)
              UpperCamelCase	:Any     							=       max(self.st[self.left(_A			)]   ,    self.st[self.right(_A			)]			)
      def    UpperCAmelCase						(    self   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_			)    ->				Union[str, Any]:
          return self.update_recursive(1   ,    0   ,    self.N - 1   ,    a - 1   ,    b - 1   ,    _A			)
      def    UpperCAmelCase						(    self   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_			)    ->				Union[str, Any]:  # noqa: E741
          if r < a or l > b:
              return True
          if l == r:
              UpperCamelCase	:Optional[int]     							=       val
              return True
          UpperCamelCase	:Union[str, Any]     							=       (l + r) // 2
          self.update_recursive(self.left(_A			)   ,    _A   ,    _A   ,    _A   ,    _A   ,    _A			)
          self.update_recursive(self.right(_A			)   ,    mid + 1   ,    _A   ,    _A   ,    _A   ,    _A			)
          UpperCamelCase	:List[str]     							=       max(self.st[self.left(_A			)]   ,    self.st[self.right(_A			)]			)
          return True
      def    UpperCAmelCase						(    self   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_			)    ->				Any:
          return self.query_recursive(1   ,    0   ,    self.N - 1   ,    a - 1   ,    b - 1			)
      def    UpperCAmelCase						(    self   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_   ,    SCREAMING_SNAKE_CASE_			)    ->				Optional[int]:  # noqa: E741
          if r < a or l > b:
              return -math.inf
          if l >= a and r <= b:
              return self.st[idx]
          UpperCamelCase	:Optional[Any]     							=       (l + r) // 2
          UpperCamelCase	:Union[str, Any]     							=       self.query_recursive(self.left(_A			)   ,    _A   ,    _A   ,    _A   ,    _A			)
          UpperCamelCase	:Tuple     							=       self.query_recursive(self.right(_A			)   ,    mid + 1   ,    _A   ,    _A   ,    _A			)
          return max(_A   ,    _A			)




      def    UpperCAmelCase						(    self			)    ->				Tuple:
          UpperCamelCase	:int     							=       []
          for i in range(1   ,    N + 1			):
              show_list += [self.query(_A   ,    _A			)]
          print(_A			)


if __name__ == "__main__":
 __snake_case							 =  [1, 2, -4, 7, 3, -5, 6, 11, -20, 9, 14, 15, 5, 2, -8]
 __snake_case							 =  15
 __snake_case							 =  SegmentTree(A)
 print(segt.query(4, 6))
 print(segt.query(7, 11))
 print(segt.query(7, 12))
 segt.update(1, 3, 1_11)
 print(segt.query(1, 15))
 segt.update(7, 8, 2_35)
 segt.show_data()













def   _A       (    SCREAMING_SNAKE_CASE__  :     list[list[int]]	):
    if len(SCREAMING_SNAKE_CASE__	) == 0 or not all(len(SCREAMING_SNAKE_CASE__	) == 2 for pair in coordinates	):
        raise ValueError('''The program cannot work out a fitting polynomial.'''	)

    if len({tuple(SCREAMING_SNAKE_CASE__	) for pair in coordinates}	) != len(SCREAMING_SNAKE_CASE__	):
        raise ValueError('''The program cannot work out a fitting polynomial.'''	)

    UpperCamelCase	:str     							=       {x for x, _ in coordinates}
    if len(SCREAMING_SNAKE_CASE__	) == 1:
        return F'''x={coordinates[0][0]}'''

    if len(SCREAMING_SNAKE_CASE__	) != len(SCREAMING_SNAKE_CASE__	):
        raise ValueError('''The program cannot work out a fitting polynomial.'''	)

    UpperCamelCase	:List[Any]     							=       len(SCREAMING_SNAKE_CASE__	)

    UpperCamelCase	:Optional[int]     							=       0
    UpperCamelCase	:list[list[float]]     							=       []
    # put the x and x to the power values in a matrix
    while count_of_line < x:
        UpperCamelCase	:Tuple     							=       0
        UpperCamelCase	:Optional[int]     							=       coordinates[count_of_line][0]
        UpperCamelCase	:list[float]     							=       []
        while count_in_line < x:
            count_line.append(a ** (x - (count_in_line + 1))	)
            count_in_line += 1
        matrix.append(SCREAMING_SNAKE_CASE__	)
        count_of_line += 1

    UpperCamelCase	:Optional[int]     							=       0
    # put the y values into a vector
    UpperCamelCase	:list[float]     							=       []
    while count_of_line < x:
        vector.append(coordinates[count_of_line][1]	)
        count_of_line += 1

    UpperCamelCase	:str     							=       0

    while count < x:
        UpperCamelCase	:Optional[Any]     							=       0
        while zahlen < x:
            if count == zahlen:
                zahlen += 1
            if zahlen == x:
                break
            UpperCamelCase	:Union[str, Any]     							=       matrix[zahlen][count] / matrix[count][count]
            for counting_columns, item in enumerate(matrix[count]	):
                # manipulating all the values in the matrix
                matrix[zahlen][counting_columns] -= item * bruch
            # manipulating the values in the vector
            vector[zahlen] -= vector[count] * bruch
            zahlen += 1
        count += 1

    UpperCamelCase	:Optional[int]     							=       0
    # make solutions
    UpperCamelCase	:list[str]     							=       []
    while count < x:
        solution.append(str(vector[count] / matrix[count][count]	)	)
        count += 1

    UpperCamelCase	:Union[str, Any]     							=       0
    UpperCamelCase	:Any     							=       '''f(x)='''

    while count < x:
        UpperCamelCase	:list[str]     							=       solution[count].split('''E'''	)
        if len(SCREAMING_SNAKE_CASE__	) > 1:
            UpperCamelCase	:List[Any]     							=       F'''{remove_e[0]}*10^{remove_e[1]}'''
        solved += F'''x^{x - (count + 1)}*{solution[count]}'''
        if count + 1 != x:
            solved += "+"
        count += 1

    return solved


if __name__ == "__main__":
 print(points_to_polynomial([]))
 print(points_to_polynomial([[]]))
 print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
 print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
 print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
 print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
 print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
 print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
 print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
 print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))





