'''simple docstring'''




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

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

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

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

    A_     =   len(_UpperCamelCase   )

    A_     =   0
    A_     =   []
    # put the x and x to the power values in a matrix
    while count_of_line < x:
        A_     =   0
        A_     =   coordinates[count_of_line][0]
        A_     =   []
        while count_in_line < x:
            count_line.append(a ** (x - (count_in_line + 1))   )
            count_in_line += 1
        matrix.append(_UpperCamelCase   )
        count_of_line += 1

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

    A_     =   0

    while count < x:
        A_     =   0
        while zahlen < x:
            if count == zahlen:
                zahlen += 1
            if zahlen == x:
                break
            A_     =   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

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

    A_     =   0
    A_     =   '''f(x)='''

    while count < x:
        A_     =   solution[count].split('''E'''   )
        if len(_UpperCamelCase   ) > 1:
            A_     =   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]]))







'''simple docstring'''




__snake_case   :      List[Any]						    =  [chr(i) for i in range(32, 126)]
__snake_case   :      Dict						    =  list(range(len(alphabets)))
__snake_case   :      Dict						    =  list(range(len(alphabets)))
__snake_case   :      Dict						    =  list(range(len(alphabets)))
__snake_case   :      List[Any]						    =  list(reversed(range(len(alphabets))))
__snake_case   :      Optional[int]						    =  []
__snake_case   :      List[Any]						    =  0





def      _UpperCAmelCase    (							)	->      Any:
    global gear_one_pos
    global gear_two_pos
    global gear_three_pos
    A_     =   gear_one[0]
    gear_one.append(_UpperCamelCase   )
    del gear_one[0]
    gear_one_pos += 1
    if gear_one_pos % int(len(_UpperCamelCase   )   ) == 0:
        A_     =   gear_two[0]
        gear_two.append(_UpperCamelCase   )
        del gear_two[0]
        gear_two_pos += 1
        if gear_two_pos % int(len(_UpperCamelCase   )   ) == 0:
            A_     =   gear_three[0]
            gear_three.append(_UpperCamelCase   )
            del gear_three[0]
            gear_three_pos += 1





def      _UpperCAmelCase    (							_UpperCamelCase    :	Tuple   )	->      Any:
    A_     =   alphabets.index(_UpperCamelCase   )
    A_     =   gear_one[target]
    A_     =   gear_two[target]
    A_     =   gear_three[target]
    A_     =   reflector[target]
    A_     =   gear_three.index(_UpperCamelCase   )
    A_     =   gear_two.index(_UpperCamelCase   )
    A_     =   gear_one.index(_UpperCamelCase   )
    code.append(alphabets[target]   )
    rotator()


if __name__ == "__main__":
  __snake_case   :      List[str]						    =  list(input('Type your message:\n'))
  while True:
    try:
      __snake_case   :      Any						    =  int(input('Please set token:(must be only digits)\n'))
      break
    except Exception as error:
      print(error)
  for _ in range(token):
    rotator()
  for j in decode:
    engine(j)
  print('\n' + ''.join(code))
  print(
      F"""\nYour Token is {token} please write it down.\nIf you want to decode """
      'this message again you should input same digits as token!'
  )






