First Passage with Periodically Varying Drift

A Markov chain on $\{0, 1, 2, 3, 4, 5, 6\}$ has absorbing state $0$ and reflecting state $6$ (i.e., $p(6, 5) = 1$). For transient states $1 \le i \le 5$, the transition probabilities depend on $i \bmod 3$:

• If $i \equiv 0 \pmod{3}$ (i.e., $i = 3$): $p(i, i-1) = \tfrac{2}{3}$, $p(i, i+1) = \tfrac{1}{3}$.
• If $i \equiv 1 \pmod{3}$ (i.e., $i \in \{1, 4\}$): $p(i, i-1) = \tfrac{1}{2}$, $p(i, i+1) = \tfrac{1}{2}$.
• If $i \equiv 2 \pmod{3}$ (i.e., $i \in \{2, 5\}$): $p(i, i-1) = \tfrac{1}{3}$, $p(i, i+1) = \tfrac{2}{3}$.

Compute $E[T_0 \mid X_0 = 3]$.