Gambler's Ruin with Killing (Absorption in the Interior)

A particle moves on $\{0, 1, 2, 3, 4, 5\}$. States $0$ and $5$ are absorbing barriers. At each step from a transient state $k$ ($1 \le k \le 4$), the particle is "killed" (absorbed into a cemetery state $\Delta$) with probability $c = 1/5$. With the remaining probability $4/5$, it moves $+1$ or $-1$ each with probability $1/2$. Starting from state $2$, find the probability that the particle reaches state $5$ (without being killed or hitting $0$).