input: 
  from typing import List
  
  class Maze:
      def __init__(self, n: int, m: int):
          self._n, self._m = n, m
          self._grid = [[None] * m for _ in range(n)]
  
      def solve_maze(self) -> List[int]:
          """Solves a maze by finding the shortest path from the start (0, 0) to the end
          (n-1, m-1) using a recursive depth-first search approach. The maze is represented
          as a grid where 1 denotes a valid path and other values denote walls.
          The function returns a list of integers representing the path length at each step.
          
          >>> maze = Maze(3, 3)
          >>> maze._grid = [[1, 0, 1], [1, 1, 1], [0, 0, 1]]
          >>> maze.solve_maze()
          [4]
          """
          result, found = [], False
          start, end = (0, 0), (self._n-1, self._m-1)
  
          for i in range(self._n):
              for j in range(self._m):
                  if self._grid[i][j] == 1:
                      value = 0
                      self._find_omega(i, j, value, result, start, end, found)
                      return result
  
      def _find_omega(self, i: int, j: int, value: int,
                      result: List[int], start: tuple, end: tuple, found: bool):
          rowNbr, colNbr = [-1, 0, 0, 1], [0, -1, 1, 0]
  
          if (i, j) == end:
              found = True
              result.append(value)
              return
  
          if found:
              return
  
          self._grid[i][j] = 0
          for k in range(4):
              if self._on_board(i + rowNbr[k], j + colNbr[k]):
                  self._find_omega(i + rowNbr[k], j + colNbr[k],
                                   value + 1, result,
                                   start, end, found)
  
      def _on_board(self, i: int, j: int) -> bool:
          return 0 <= i < self._n and 0 <= j < self._m and self._grid[i][j] == 1
  
output:     
          if not found:
              self._grid[i][j] = 1
  