


from bisect import bisect
from itertools import accumulate
def 				UpperCamelCase__					(  SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						):
 __lowerCamelCase		:							int		 =						sorted(zip(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)      ,	key=lambda SCREAMING_SNAKE_CASE__						: x[0] / x[1]      ,	reverse=SCREAMING_SNAKE_CASE__						)
 __lowerCamelCase		:							Dict		 =						[i[0] for i in r], [i[1] for i in r]
 __lowerCamelCase		:							Optional[Any]		 =						list(accumulate(SCREAMING_SNAKE_CASE__						)						)
 __lowerCamelCase		:							Union[str, Any]		 =						bisect(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 return (
     0
     if k == 0
     else sum(vl[:k]						) + (w - acc[k - 1]) * (vl[k]) / (wt[k])
     if k != n
     else sum(vl[:k]						)
 )


if __name__ == "__main__":
       import doctest

       doctest.testmod()





from math import pow
def 				UpperCamelCase__					(  SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	):
 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

 __lowerCamelCase		:							Optional[Any]		 =						int(pow(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)						)
 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
  __lowerCamelCase ,					__lowerCamelCase		:							Optional[Any]		 =						backtrack(
      SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	current_number + 1      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
  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.
  __lowerCamelCase ,					__lowerCamelCase		:							Dict		 =						backtrack(
      SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	current_number + 1      ,	SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						)
 return current_sum, solutions_count
def 				UpperCamelCase__					(  SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__						):
 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(SCREAMING_SNAKE_CASE__      ,	SCREAMING_SNAKE_CASE__      ,	1      ,	0      ,	0						)[1]  # Return the solutions_count


if __name__ == "__main__":
       import doctest

       doctest.testmod()

