'''simple docstring'''

from __future__ import annotations
def 			__UpperCamelCase       (   UpperCAmelCase     ):
 if len(UpperCAmelCase     ) == 0:
  return []
 lowercase__ ,							lowercase__				:    List[Any]  =					min(UpperCAmelCase     ), max(UpperCAmelCase     )
 lowercase__				:    List[str]  =					int(max_value - min_value     ) + 1
 lowercase__				:    list[list]  =					[[] for _ in range(UpperCAmelCase     )]

 for i in my_list:
  buckets[int(i - min_value     )].append(UpperCAmelCase     )

 return [v for bucket in buckets for v in sorted(UpperCAmelCase     )]


if __name__ == "__main__":
  from doctest import testmod

  testmod()
  assert bucket_sort([4, 5, 3, 2, 1]) == [1, 2, 3, 4, 5]
  assert bucket_sort([0, 1, -10, 15, 2, -2]) == [-10, -2, 0, 1, 2, 15]


'''simple docstring'''

from math import pow
def 			__UpperCamelCase       (   UpperCAmelCase		,     UpperCAmelCase		,     UpperCAmelCase		,     UpperCAmelCase		,     UpperCAmelCase		,     ):
 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

 lowercase__				:    Optional[Any]  =					int(pow(UpperCAmelCase		,     UpperCAmelCase     )     )
 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
  lowercase__ ,							lowercase__				:    Dict  =					backtrack(
      UpperCAmelCase		,     UpperCAmelCase		,     current_number + 1		,     UpperCAmelCase		,     UpperCAmelCase     )
  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.
  lowercase__ ,							lowercase__				:    str  =					backtrack(
      UpperCAmelCase		,     UpperCAmelCase		,     current_number + 1		,     UpperCAmelCase		,     UpperCAmelCase     )
 return current_sum, solutions_count
def 			__UpperCamelCase       (   UpperCAmelCase		,     UpperCAmelCase     ):
 if not (1 <= needed_sum <= 1000 and 2 <= power <= 10):
  raise ValueError(
      '''Invalid input\n'''
      '''needed_sum must be between 1 and 1000, power between 2 and 10.'''     )

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


if __name__ == "__main__":
  import doctest

  doctest.testmod()

