Markov Chains & Long-Run Behavior · Superchallenge problem

A robot does a random walk on a $3 \times 3$ grid of squares, each step moving to a uniformly chosen square that shares a side with its current square. It starts in a corner square. Let $q_n$ be the probability that it is on the centre square after exactly $n$ steps. What is $\displaystyle\lim_{n \to \infty} q_{2n}$? Express your answer as a common fraction.