Markov Chains & Long-Run Behavior · Hard problem

A particle sits at position $1$ of $\{0, 1, 2\}$. From $1$ it moves to $0$ or to $2$, each with probability $1/2$; from $0$ it always moves to $1$, and from $2$ it always moves to $1$. What is the expected number of steps until the particle has visited BOTH $0$ and $2$ at least once?