





from math import pow


def     SCREAMING_SNAKE_CASE			(      lowerCAmelCase    ,       lowerCAmelCase    ,       lowerCAmelCase    ,       lowerCAmelCase    ,       lowerCAmelCase    ,       ):
							if current_sum == needed_sum:
														# If the sum of the powers is equal to needed_sum, then we have a solution.
														solutions_count += 1
														return current_sum, solutions_count

							_UpperCamelCase              =						int(pow(lowerCAmelCase    ,       lowerCAmelCase					)					)
							if current_sum + i_to_n <= needed_sum:
														# If the sum of the powers is less than needed_sum, then continue adding powers.
														current_sum += i_to_n
														_UpperCamelCase       ,		_UpperCamelCase              =						backtrack(
														    lowerCAmelCase    ,       lowerCAmelCase    ,       current_number + 1    ,       lowerCAmelCase    ,       lowerCAmelCase					)
														current_sum -= i_to_n
							if i_to_n < needed_sum:
														# If the power of i is less than needed_sum, then try with the next power.
														_UpperCamelCase       ,		_UpperCamelCase              =						backtrack(
														    lowerCAmelCase    ,       lowerCAmelCase    ,       current_number + 1    ,       lowerCAmelCase    ,       lowerCAmelCase					)
							return current_sum, solutions_count


def     SCREAMING_SNAKE_CASE			(      lowerCAmelCase    ,       lowerCAmelCase					):
							if not (1 <= needed_sum <= 1_000 and 2 <= power <= 10):
														raise ValueError(
														    '''Invalid input\n'''
														    '''needed_sum must be between 1 and 1000, power between 2 and 10.'''					)

							return backtrack(lowerCAmelCase    ,       lowerCAmelCase    ,       1    ,       0    ,       0					)[1]  # Return the solutions_count


if __name__ == "__main__":
							import doctest

							doctest.testmod()













def     SCREAMING_SNAKE_CASE			(      lowerCAmelCase					):
       _UpperCamelCase              =						[]
       if len(lowerCAmelCase					) == 1:
              return [nums.copy()]
       for _ in range(len(lowerCAmelCase					)					):
              _UpperCamelCase              =						nums.pop(0					)
              _UpperCamelCase              =						permute(lowerCAmelCase					)
              for perm in permutations:
                     perm.append(lowerCAmelCase					)
              result.extend(lowerCAmelCase					)
              nums.append(lowerCAmelCase					)
       return result


def     SCREAMING_SNAKE_CASE			(      lowerCAmelCase					):

       def backtrack(lowerCAmelCase					):
              if start == len(lowerCAmelCase					) - 1:
                     output.append(nums[:]					)
              else:
                     for i in range(lowerCAmelCase    ,       len(lowerCAmelCase					)					):
                            _UpperCamelCase       ,		_UpperCamelCase              =						nums[i], nums[start]
                            backtrack(start + 1					)
                            _UpperCamelCase       ,		_UpperCamelCase              =						nums[i], nums[start]  # backtrack

       _UpperCamelCase              =						[]
       backtrack(0					)
       return output


if __name__ == "__main__":
       import doctest

       # use res to print the data in permute2 function
       lowercase    : Optional[Any]    	=  permutea([1, 2, 3])
       print(res)
       doctest.testmod()






