Gambler's Ruin with a Partially Reflecting Barrier

Consider a Markov chain on $\{0, 1, 2, \ldots\}$ with absorbing state $0$. From state $k \ge 1$, the chain moves to $k+1$ with probability $p$ and to $k-1$ with probability $q = 1 - p$, where $0 < p < 1$. However, there is a reflecting barrier at state $N$: from state $N$, the chain moves to $N-1$ with probability $1$ (it is always pushed back).

Starting from state $k$ with $1 \le k \le N$:

(a) Find the probability $r_k$ of eventual absorption at $0$.

(b) For $p = q = \tfrac{1}{2}$ and $N = 4$, compute $r_2$ and the expected absorption time $E[T \mid X_0 = 2]$.