First Passage Through a Randomizing State

A Markov chain on $\{0, 1, 2\}$ has the transition matrix $$P = \begin{pmatrix} 1 & 0 & 0 \\ \tfrac{1}{4} & 0 & \tfrac{3}{4} \\ \tfrac{1}{2} & \tfrac{1}{2} & 0 \end{pmatrix}.$$ State $0$ is absorbing. Starting from state $1$, let $T = \inf\{n \ge 1 : X_n = 0\}$.

(a) Compute $E[T \mid X_0 = 1]$.

(b) Compute $E[T \mid X_0 = 2]$.