Markov Chains & Long-Run Behavior · Challenge problem

A symmetric two-state chain has transition matrix $M = \begin{pmatrix} a & 1-a \\ 1-a & a \end{pmatrix}$ with $a > 1/2$. The two-step matrix is $M^2 = \begin{pmatrix} 5/8 & 3/8 \\ 3/8 & 5/8 \end{pmatrix}$. What is $a$? Express your answer as a common fraction.