In chess, a bishop is allowed to move only in straight diagonal lines. Thus if the bishop stands on the square marked $\mathrm{A}$ in the diagram, it is able in one move to reach any of the squares marked with an asterisk. Suppose that the bishop moves at random around the chess board, choosing at each move with equal probability from the squares it can reach, the square chosen being independent of all previous choices. The bishop starts at the bottom left-hand corner of the board.

If $X_{n}$ is the position of the bishop at time $n$, show that $\left(X_{n}\right)_{n \geqslant 0}$ is a reversible Markov chain, whose statespace you should specify. Find the invariant distribution of this Markov chain.

What is the expected number of moves the bishop will make before first returning to its starting square?