First Passage in a Three-State Chain with Self-Loop

A Markov chain on $\{0, 1, 2\}$ has transition matrix $$P = \begin{pmatrix} 1 & 0 & 0 \\ \tfrac{1}{5} & \tfrac{2}{5} & \tfrac{2}{5} \\ 0 & \tfrac{3}{4} & \tfrac{1}{4} \end{pmatrix}.$$ State $0$ is absorbing. Compute $E[T_0 \mid X_0 = 1]$, where $T_0 = \inf\{n \ge 1 : X_n = 0\}$.