




'''simple docstring'''






def       __lowerCAmelCase (   snake_case__						):
   if len(snake_case__						) == 0 or not all(len(snake_case__						) == 2 for pair in coordinates						):
      raise ValueError("The program cannot work out a fitting polynomial."						)

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

   __UpperCamelCase				:      Tuple									=       {x for x, _ in coordinates}
   if len(snake_case__						) == 1:
      return F"x={coordinates[0][0]}"

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

   __UpperCamelCase				:      str									=       len(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				:      int									=       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(snake_case__						)
      count_of_line += 1

   __UpperCamelCase				:      Any									=       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				:      Union[str, Any]									=       0

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

   __UpperCamelCase				:      Any									=       0
   __UpperCamelCase				:      Dict									=       'f(x)='

   while count < x:
      __UpperCamelCase				:      list[str]									=       solution[count].split("E"						)
      if len(snake_case__						) > 1:
         __UpperCamelCase				:      Union[str, 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]]))









'''simple docstring'''






def       __lowerCAmelCase (   snake_case__ = 4_000_000						):
			__UpperCamelCase				:      Tuple									=       1
			__UpperCamelCase				:      int									=       2
			__UpperCamelCase				:      str									=       0
			while j <= n:
						if j % 2 == 0:
									total += j
						__UpperCamelCase		, __UpperCamelCase				:      List[Any]									=       j, i + j

			return total


if __name__ == "__main__":
				print(f'{solution() = }')



