Four-state reflecting absorbing random walk

For the chain with $0$ absorbing, equal-probability nearest-neighbour moves between $1,2,3$, and a probability-$1/2$ self-loop at $3$, starting from $1$ the expected absorption time is $6$, the probability of visiting $3$ before $0$ is $1/3$, and the expected number of visits to $3$ before absorption is $2$.