Three-State First Passage with Deterministic Return

A Markov chain on $\{0, 1, 2\}$ has transition probabilities: $$p(0,1) = 1, \quad p(1,0) = \tfrac{2}{5}, \quad p(1,2) = \tfrac{3}{5}, \quad p(2,2) = 1.$$ Compute $E[T_2 \mid X_0 = 0]$, where $T_2 = \inf\{n \ge 0 : X_n = 2\}$.