






def     _snake_case    (__lowercase    ,      __lowercase):
   UpperCamelCase_									=     word.split()

   def justify(__lowercase    ,      __lowercase    ,      __lowercase) -> str:
      UpperCamelCase_									=     max_width - width
      UpperCamelCase_									=     len(__lowercase)
      if len(__lowercase) == 1:
         # if there is only word in line
         # just insert overall_spaces_count for the remainder of line
         return line[0] + " " * overall_spaces_count
      else:
         UpperCamelCase_									=     words_count - 1
         # num_spaces_between_words_list[i] : tells you to insert
         # num_spaces_between_words_list[i] spaces
         # after word on line[i]
         UpperCamelCase_									=     spaces_to_insert_between_words * [
             overall_spaces_count // spaces_to_insert_between_words
         ]
         UpperCamelCase_									=     (
             overall_spaces_count % spaces_to_insert_between_words
         )
         # distribute spaces via round robin to the left words
         for i in range(__lowercase):
            num_spaces_between_words_list[i] += 1
         UpperCamelCase_									=     []
         for i in range(__lowercase):
            # add the word
            aligned_words_list.append(line[i])
            # add the spaces to insert
            aligned_words_list.append(num_spaces_between_words_list[i] * ' ')
         # just add the last word to the sentence
         aligned_words_list.append(line[-1])
         # join the aligned words list to form a justified line
         return "".join(__lowercase)

   UpperCamelCase_									=     []
   UpperCamelCase_									=     []
   UpperCamelCase_									=     0
   for word in words:
      if width + len(__lowercase) + len(__lowercase) <= max_width:
         # keep adding words until we can fill out max_width
         # width = sum of length of all words (without overall_spaces_count)
         # len(word) = length of current word
         # len(line) = number of overall_spaces_count to insert between words
         line.append(__lowercase)
         width += len(__lowercase)
      else:
         # justify the line and add it to result
         answer.append(justify(__lowercase    ,      __lowercase    ,      __lowercase))
         # reset new line and new width
         UpperCamelCase_		,      UpperCamelCase_									=     [word], len(__lowercase)
   UpperCamelCase_									=     max_width - width - len(__lowercase)
   answer.append(' '.join(__lowercase) + (remaining_spaces + 1) * ' ')
   return answer


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

 testmod()













def     _snake_case    (__lowercase    ,      __lowercase):
   _enforce_args(__lowercase    ,      __lowercase)
   if n == 0:
      return 0
   UpperCamelCase_									=     float('-inf')
   for i in range(1    ,      n + 1):
      UpperCamelCase_									=     max(
          __lowercase    ,      prices[i - 1] + naive_cut_rod_recursive(n - i    ,      __lowercase))

   return max_revue






def     _snake_case    (__lowercase    ,      __lowercase):
   _enforce_args(__lowercase    ,      __lowercase)
   UpperCamelCase_									=     [float('-inf') for _ in range(n + 1)]
   return _top_down_cut_rod_recursive(__lowercase    ,      __lowercase    ,      __lowercase)






def     _snake_case    (__lowercase    ,      __lowercase    ,      __lowercase):
   if max_rev[n] >= 0:
      return max_rev[n]
   elif n == 0:
      return 0
   else:
      UpperCamelCase_									=     float('-inf')
      for i in range(1    ,      n + 1):
         UpperCamelCase_									=     max(
             __lowercase    ,      prices[i - 1] + _top_down_cut_rod_recursive(n - i    ,      __lowercase    ,      __lowercase)    ,      )

      UpperCamelCase_									=     max_revenue

   return max_rev[n]






def     _snake_case    (__lowercase    ,      __lowercase):
   _enforce_args(__lowercase    ,      __lowercase)

   # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
   # length 0.
   UpperCamelCase_									=     [float('-inf') for _ in range(n + 1)]
   UpperCamelCase_									=     0

   for i in range(1    ,      n + 1):
      UpperCamelCase_									=     max_rev[i]
      for j in range(1    ,      i + 1):
         UpperCamelCase_									=     max(__lowercase    ,      prices[j - 1] + max_rev[i - j])

      UpperCamelCase_									=     max_revenue_i

   return max_rev[n]






def     _snake_case    (__lowercase    ,      __lowercase):
   if n < 0:
      UpperCamelCase_									=     f"""n must be greater than or equal to 0. Got n = {n}"""
      raise ValueError(__lowercase)

   if n > len(__lowercase):
      UpperCamelCase_									=     (
          'Each integral piece of rod must have a corresponding price. '
          f"""Got n = {n} but length of prices = {len(__lowercase)}"""
      )
      raise ValueError(__lowercase)






def     _snake_case    ():
   UpperCamelCase_									=     [6, 10, 12, 15, 20, 23]
   UpperCamelCase_									=     len(__lowercase)

   # the best revenue comes from cutting the rod into 6 pieces, each
   # of length 1 resulting in a revenue of 6 * 6 = 36.
   UpperCamelCase_									=     36

   UpperCamelCase_									=     top_down_cut_rod(__lowercase    ,      __lowercase)
   UpperCamelCase_									=     bottom_up_cut_rod(__lowercase    ,      __lowercase)
   UpperCamelCase_									=     naive_cut_rod_recursive(__lowercase    ,      __lowercase)

   assert expected_max_revenue == max_rev_top_down
   assert max_rev_top_down == max_rev_bottom_up
   assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
 main()





