Hitting Time on a Product of Two-State Chains

Let $(X_n, Y_n)$ be a Markov chain on $\{0,1\}^2$ where $X_n$ and $Y_n$ evolve independently. At each step, $X_n$ flips (i.e., $X_{n+1} = 1 - X_n$) with probability $\tfrac{1}{3}$ and stays with probability $\tfrac{2}{3}$. Similarly, $Y_n$ flips with probability $\tfrac{1}{2}$ and stays with probability $\tfrac{1}{2}$.

Starting from $(X_0, Y_0) = (1, 1)$, compute the expected number of steps to first reach the state $(0, 0)$.