Expected tosses until three identical faces in a row

Two players toss a fair coin over and over. Anna wins as soon as three consecutive heads occur, and Brenda wins as soon as three consecutive tails occur. On average, how many tosses does it take to decide the winner?

$7$. After the first toss we are in a run of length $1$. If $\mu_1,\mu_2$ are expected remaining tosses from runs of length $1$ and $2$, then $\mu_1=1+\tfrac12\mu_2+\tfrac12\mu_1$ and $\mu_2=1+\tfrac12\mu_1$, giving $\mu_1=6$; adding the first toss gives $7$.