Simplify absorbing Markov chain

Question

I am applying a discrete-time Markov model to a board game, to find the expected number of turns until it reaches the absorbing state. I understand that I must find the fundamental matrix to do this. The issue is that there are many millions of states ($(4^{13})$. Is there a way to simplify this? It is worth noting that it is a sparse matrix, and there are 9 communication classes, where one of those classes is the final absorbing state. Each of the other 8 communication classes has the same number of states within them. States transition from each class in order i.e. communication class 1 can go to class 2,3,4... but class 2 cannot go back to class 1, class 3 cannot go to 1 or 2, etc. Is there a way to simplify this.