





def 	_SCREAMING_SNAKE_CASE		(      SCREAMING_SNAKE_CASE		:list[list]   )      ->   list[list]:
 __lowerCAmelCase					:	Optional[int]      =       current_set.copy()
 for row_index, row in enumerate(lowerCAmelCase__   ):
  __lowerCAmelCase					:	Any      =       row[0]
  for column_index, column in enumerate(lowerCAmelCase__   ):
   if magnitude == 0:
    __lowerCAmelCase					:	Optional[Any]      =       column
    continue
   __lowerCAmelCase					:	Optional[Any]      =       column / magnitude
    # Subtract to cancel term
 __lowerCAmelCase					:	Optional[int]      =       current_set[0]
 __lowerCAmelCase					:	List[str]      =       [first_row]
 __lowerCAmelCase					:	Any      =       current_set[1::]
 for row in current_set:
  __lowerCAmelCase					:	Optional[int]      =       []
  # If first term is 0, it is already in form we want, so we preserve it
  if row[0] == 0:
   final_set.append(lowerCAmelCase__   )
   continue
  for column_index in range(len(lowerCAmelCase__   )   ):
   temp_row.append(first_row[column_index] - row[column_index]   )
  final_set.append(lowerCAmelCase__   )
 # Create next recursion iteration set
 if len(final_set[0]   ) != 3:
  __lowerCAmelCase					:	List[str]      =       final_set[0]
  __lowerCAmelCase					:	str      =       []
  __lowerCAmelCase					:	int      =       []
  for row in final_set[1::]:
   current_first_column.append(row[0]   )
   next_iteration.append(row[1::]   )
  __lowerCAmelCase					:	Optional[Any]      =       simplify(lowerCAmelCase__   )
  for i in range(len(lowerCAmelCase__   )   ):
   resultant[i].insert(0   ,   current_first_column[i]   )
  resultant.insert(0   ,   lowerCAmelCase__   )
  __lowerCAmelCase					:	Tuple      =       resultant
 return final_set






def 	_SCREAMING_SNAKE_CASE		(      SCREAMING_SNAKE_CASE		:list[list]   )      ->   list:


 if len(lowerCAmelCase__   ) == 0:
  raise IndexError("""solve_simultaneous() requires n lists of length n+1"""   )
 __lowerCAmelCase					:	Optional[Any]      =       len(lowerCAmelCase__   ) + 1
 if any(len(lowerCAmelCase__   ) != _length for item in equations   ):
  raise IndexError("""solve_simultaneous() requires n lists of length n+1"""   )
 for row in equations:
  if any(not isinstance(lowerCAmelCase__   ,   (int, float)   ) for column in row   ):
   raise ValueError("""solve_simultaneous() requires lists of integers"""   )
 if len(lowerCAmelCase__   ) == 1:
  return [equations[0][-1] / equations[0][0]]
 __lowerCAmelCase					:	Dict      =       equations.copy()
 if any(0 in row for row in data_set   ):
  __lowerCAmelCase					:	Optional[Any]      =       data_set.copy()
  __lowerCAmelCase					:	Optional[int]      =       []
  for row_index, row in enumerate(lowerCAmelCase__   ):
   if 0 not in row:
    __lowerCAmelCase					:	Optional[int]      =       data_set.pop(lowerCAmelCase__   )
    break
  if not full_row:
   raise ValueError("""solve_simultaneous() requires at least 1 full equation"""   )
  data_set.insert(0   ,   lowerCAmelCase__   )
 __lowerCAmelCase					:	str      =       data_set.copy()
 __lowerCAmelCase					:	int      =       simplify(lowerCAmelCase__   )
 __lowerCAmelCase					:	List[Any]      =       simplified[::-1]
 __lowerCAmelCase					:	Tuple      =       []
 for row in simplified:
  __lowerCAmelCase					:	Optional[int]      =       row[-1]
  if not solutions:
   if row[-2] == 0:
    solutions.append(0   )
    continue
   solutions.append(current_solution / row[-2]   )
   continue
  __lowerCAmelCase					:	List[str]      =       row.copy()[: len(lowerCAmelCase__   ) - 1 :]
  while temp_row[0] == 0:
   temp_row.pop(0   )
  if len(lowerCAmelCase__   ) == 0:
   solutions.append(0   )
   continue
  __lowerCAmelCase					:	Tuple      =       temp_row[1::]
  __lowerCAmelCase					:	int      =       temp_row[::-1]
  for column_index, column in enumerate(lowerCAmelCase__   ):
   current_solution -= column * solutions[column_index]
  solutions.append(lowerCAmelCase__   )
 __lowerCAmelCase					:	str      =       []
 for item in solutions:
  final.append(float(round(lowerCAmelCase__   ,   5   )   )   )
 return final[::-1]


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      A             =							[
          [2, 1, 1, 1, 1, 4],
          [1, 2, 1, 1, 1, 5],
          [1, 1, 2, 1, 1, 6],
          [1, 1, 1, 2, 1, 7],
          [1, 1, 1, 1, 2, 8],
      ]
      print(solve_simultaneous(eq))
      print(solve_simultaneous([[4, 2]]))






def 	_SCREAMING_SNAKE_CASE		(      SCREAMING_SNAKE_CASE		:int   ,   SCREAMING_SNAKE_CASE		:list   )      ->   Dict:
 _enforce_args(SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   )
 if n == 0:
  return 0
 __lowerCAmelCase					:	Union[str, Any]      =       float("""-inf"""   )
 for i in range(1   ,   n + 1   ):
  __lowerCAmelCase					:	Union[str, Any]      =       max(
      SCREAMING_SNAKE_CASE   ,   prices[i - 1] + naive_cut_rod_recursive(n - i   ,   SCREAMING_SNAKE_CASE   )   )

 return max_revue






def 	_SCREAMING_SNAKE_CASE		(      SCREAMING_SNAKE_CASE		:int   ,   SCREAMING_SNAKE_CASE		:list   )      ->   List[str]:
 _enforce_args(SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   )
 __lowerCAmelCase					:	List[str]      =       [float("""-inf"""   ) for _ in range(n + 1   )]
 return _top_down_cut_rod_recursive(SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   )






def 	_SCREAMING_SNAKE_CASE		(      SCREAMING_SNAKE_CASE		:int   ,   SCREAMING_SNAKE_CASE		:list   ,   SCREAMING_SNAKE_CASE		:list   )      ->   List[str]:
 if max_rev[n] >= 0:
  return max_rev[n]
 elif n == 0:
  return 0
 else:
  __lowerCAmelCase					:	str      =       float("""-inf"""   )
  for i in range(1   ,   n + 1   ):
   __lowerCAmelCase					:	List[Any]      =       max(
       SCREAMING_SNAKE_CASE   ,   prices[i - 1] + _top_down_cut_rod_recursive(n - i   ,   SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   )   ,   )

  __lowerCAmelCase					:	List[str]      =       max_revenue

 return max_rev[n]






def 	_SCREAMING_SNAKE_CASE		(      SCREAMING_SNAKE_CASE		:int   ,   SCREAMING_SNAKE_CASE		:list   )      ->   Union[str, Any]:
 _enforce_args(SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   )

 # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
 # length 0.
 __lowerCAmelCase					:	Optional[int]      =       [float("""-inf"""   ) for _ in range(n + 1   )]
 __lowerCAmelCase					:	List[str]      =       0

 for i in range(1   ,   n + 1   ):
  __lowerCAmelCase					:	Optional[Any]      =       max_rev[i]
  for j in range(1   ,   i + 1   ):
   __lowerCAmelCase					:	Optional[int]      =       max(SCREAMING_SNAKE_CASE   ,   prices[j - 1] + max_rev[i - j]   )

  __lowerCAmelCase					:	Optional[int]      =       max_revenue_i

 return max_rev[n]






def 	_SCREAMING_SNAKE_CASE		(      SCREAMING_SNAKE_CASE		:int   ,   SCREAMING_SNAKE_CASE		:list   )      ->   List[Any]:
 if n < 0:
  __lowerCAmelCase					:	Any      =       F'''n must be greater than or equal to 0. Got n = {n}'''
  raise ValueError(SCREAMING_SNAKE_CASE   )

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






def 	_SCREAMING_SNAKE_CASE		(      )      ->   Optional[int]:
 __lowerCAmelCase					:	Tuple      =       [6, 10, 12, 15, 20, 23]
 __lowerCAmelCase					:	List[str]      =       len(SCREAMING_SNAKE_CASE   )

 # the best revenue comes from cutting the rod into 6 pieces, each
 # of length 1 resulting in a revenue of 6 * 6 = 36.
 __lowerCAmelCase					:	Union[str, Any]      =       36

 __lowerCAmelCase					:	Optional[int]      =       top_down_cut_rod(SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   )
 __lowerCAmelCase					:	List[str]      =       bottom_up_cut_rod(SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   )
 __lowerCAmelCase					:	int      =       naive_cut_rod_recursive(SCREAMING_SNAKE_CASE   ,   SCREAMING_SNAKE_CASE   )

 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()