Expected Time to Visit All States in an Ergodic Chain

A Markov chain on $\{1, 2, 3, 4\}$ has transition matrix $$P = \begin{pmatrix} 0 & 1 & 0 & 0 \\ \tfrac{1}{3} & 0 & \tfrac{2}{3} & 0 \\ 0 & \tfrac{1}{2} & 0 & \tfrac{1}{2} \\ 0 & 0 & \tfrac{1}{2} & \tfrac{1}{2} \end{pmatrix}.$$

Starting from state $1$, let $T_{\text{cover}} = \inf\{n \ge 0 : \{1,2,3,4\} \subseteq \{X_0, X_1, \ldots, X_n\}\}$ be the first time all four states have been visited.

Compute $E[T_{\text{cover}} \mid X_0 = 1]$.

*Hint:* Decompose the cover time by tracking which states remain unvisited. Since the chain starts at state 1 and must go to 2, then eventually reach 3 and 4, compute the expected time to reach each new state sequentially.