Hitting Time with Harmonically Increasing Forward Probability

A Markov chain on $\{0, 1, 2, 3\}$ has transitions: from state $i$ ($0 \le i \le 2$), move to $i+1$ with probability $\frac{i+1}{i+2}$ and stay at $i$ with probability $\frac{1}{i+2}$. State $3$ is absorbing.

Compute $E[T_3 \mid X_0 = 0]$.