Which subway stop do you visit last?

A subway line is a loop with $n$ stations arranged in a circle. A rider starts at station $0$ and, at each step, moves to the neighbouring station clockwise or counter-clockwise with probability $\tfrac12$ each. He keeps wandering until he has set foot in every station at least once.

Show that each of the other $n-1$ stations is equally likely to be the last new station he visits, probability $\tfrac{1}{n-1}$ each.