Markov strong property exercise

Let $P=(p_{ij})$ and let $I$ be the state space. Suppose $X_0=i\not\in A$, so $T\geq1$. By the usual Markov property we have $$\mathbb P_i(T<\infty\mid X_1=j)=\mathbb P_j(T<\infty)=u(j).$$ Using the law of total probability, we can condition on the possible values of $X_1$ to get \begin{align*} u(i)=\mathbb P_i(T<\infty)=\sum_{j\in I}\mathbb P_i(T<\infty\mid X_1=j)\cdot\mathbb P_i(X_1=j)=\sum_{j\in I}p_{ij}\cdot u(j)=(Pu)(i). \end{align*}

The starting point is that if $x \not \in A$ then $P(T<\infty \mid X_0=x) =\sum_y P(T<\infty \mid X_0=x,X_1=y) P(X_1=y \mid X_0=x)$. This follows from using the total probability formula via conditioning on the outcome of the first step. Then you need to simplify that using the other properties of the situation.