Hitting Time on a Six-State Chain with Paired Coordinates

A Markov chain lives on $S = \{0, 1, 2\} \times \{0, 1\}$. The state $(0, y)$ for any $y$ is absorbing. From state $(i, j)$ with $i \ge 1$, the transitions are:

• Move to $(i-1, j)$ with probability $\tfrac{1}{3}$.
• Move to $(i, 1-j)$ with probability $\tfrac{1}{3}$ (flip the second coordinate).
• Move to $(i+1, j)$ with probability $\tfrac{1}{3}$, provided $i+1 \le 2$; if $i = 2$, this probability is redistributed equally to the other two moves.

Let $A = \{(0, 0), (0, 1)\}$ and $T = \inf\{n \ge 0 : X_n \in A\}$. Compute $E[T \mid X_0 = (1, 1)]$.