Basics: robot on a walled grid (state + step loop)

A robot stands in a room of width × height cells. Cells are (x, y) with 0 <= x < width and 0 <= y < height. The robot starts at (0, 0) facing north (north is y + 1, east is x + 1).

Write walk(width: int, height: int, commands: str) -> tuple[int, int, str] that runs the commands in order and returns the final (x, y, facing), where facing is one of "N", "E", "S", "W":

• "L": turn 90° left, staying on the same cell.
• "R": turn 90° right, staying on the same cell.
• "F": move one cell forward. If that cell would be outside the room, the robot bumps the wall and stays where it is (it keeps its facing).

walk(3, 3, "FFRFF")    # (2, 2, "E")
walk(3, 3, "FFFFF")    # (0, 2, "N")   bumps the north wall three times
walk(2, 2, "L")        # (0, 0, "W")

• 1 <= width, height <= 100; 0 <= len(commands) <= 10^4; commands only contain L, R, F.