Markov Chains & Long-Run Behavior · Superchallenge problem

A walker on the eight vertices of a regular octagon steps to one of the two neighbouring vertices with probability $1/2$ each. It starts at vertex $V$. Independently, a target vertex $T$ is chosen uniformly at random from all eight vertices (possibly $T = V$). What is the expected number of steps until the walker first stands on $T$ (zero if $T = V$)? Express your answer as a common fraction.