I have a continuous time Markov chain with Q-matrix
$$\begin{bmatrix} -3 & 2 & 0 & 0 &1\\ 0 & -3 & 3 & 0 & 0 \\ 0 & 5 & -5 & 0 & 0\\ 0 & 0 & 0 & -2 & 2\\ 0 & 0 & 0 & -1 & 1 \end{bmatrix}$$
and corresponding jump chain matrix
$$\begin{bmatrix} 0 & \frac{2}{3} & 0 & 0 &\frac{1}{3}\\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1 & 0 \end{bmatrix}$$
How do I work out the expected time to hit 4 starting from 1?
I've tried conditioning on the first step, but the first step can potentially get stuck in a loop which never hits 4 so the law of total probability for expectations results in an infinite time to hit 4.