First Passage with Parity-Dependent Transitions

Consider a Markov chain on $\{0, 1, 2, 3\}$ where states $0$ and $3$ are absorbing. The transition probabilities for transient states depend on the parity of the state:

• From any odd state $i$: $p(i, i-1) = \tfrac{3}{4}$, $p(i, i+1) = \tfrac{1}{4}$.
• From any even transient state $i$ (i.e., $i = 2$): $p(i, i-1) = \tfrac{1}{4}$, $p(i, i+1) = \tfrac{3}{4}$.

Compute $E[T \mid X_0 = 1]$ where $T = \inf\{n \ge 0 : X_n \in \{0, 3\}\}$.