DSE 2016 Question 32: Probability and Bayes' theorem

Setup, from Q31 of the paper:

Four cities A, B, C, D sit at the vertices of a square ABCD, connected by roads forming the four sides. If Mr. Rand Walker is at city $i$ in period $t$, then in period $t+1$ he moves, with probability $\tfrac12$ each, to one of the two vertices adjacent to $i$. If he is at city A at time $t=0$, the respective probabilities that he is at A, B, C, D in period $t=10$ are:

(Same random walk on the square as the previous question, starting at city A at $t=0$.) What is the probability that Mr. Walker never visits city A again up to and including period $t=10$?