Simulate one execution of the instructions. Track the robot's position (x, y) and direction.
After one cycle, the robot is bounded if either:
It returned to the origin (0, 0). It will repeat the same loop forever.
It's NOT facing north. If it faces east or west, after cycles the displacement vectors cancel out. If it faces south, after cycles they cancel. In all these cases the robot traces a closed path.
The only unbounded case is when the robot ends at a non-origin position AND still faces north. Then each cycle adds the same displacement, drifting forever.