Expected Value, Linearity & Advanced Probability · Hard problem

A mouse starts in room $A$. From $A$ it exits with probability $\tfrac12$, else moves to room $B$. From $B$ it exits with probability $\tfrac13$, else returns to $A$. Each move takes one minute. Find the expected number of minutes until the mouse exits, starting from $A$.

The answer is a fraction $\tfrac{m}{n}$ in lowest terms; find $m + n$.