Cover Time of the Path Graph P₃

A random walk moves on the path graph $P_3$ with vertices $\{1, 2, 3\}$ and edges $\{1{-}2, 2{-}3\}$. At each step, the walker moves to a uniformly random neighbor (so from vertex $2$ it goes to $1$ or $3$ each with probability $\tfrac{1}{2}$, and from vertex $1$ or $3$ it moves deterministically to $2$). Starting at vertex $1$, what is the expected number of steps to visit all three vertices (the cover time)?