Suppose that $(X_n)_{n\geq1}$ is markov chain with state space $S=\{A,B,C,D,E \}$ with the following transition matrix $$ P = \left( \begin{matrix} 0.6 & 0.4 & 0 & 0 & 0 \\ 0.3 & 0.7 & 0 & 0 & 0 \\ 0.2 & 0 & 0.4 & 0 & 0.4 \\ 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} \right) $$
I want to find the following probability $\lim_nP(X_n=A|X_0=C)$
I found that $C_A=\{A,B\}$ is a communicating class with both $A,B$ recurrent states. There is only one way to get from state $C$ to state $A$ and that happens with probability $p_{AB}=0.2$. I also know that once we hit the set $C_A$ this can be seen as an irreducible markov chain on state space ${A,B}$ and the one step probabilities will converge to $\pi_A=\frac{3}{7},\pi_B=\frac{4}{7}$. How to combine this to find $\lim_nP(X_n=A|X_0=C)$?