Markov Chains & Long-Run Behavior · Challenge problem

A chain on states $A$, $B$, $C$ moves as follows. From $A$ it goes to $B$ or to $C$ with probability $1/2$ each. From $B$ it goes to $A$, stays at $B$, or goes to $C$, each with probability $1/3$. From $C$ it always goes to $A$. Starting at $A$, what is the expected number of visits to $B$ before the chain first returns to $A$? Express your answer as a common fraction.