Markov Chains & Long-Run Behavior · Hard problem

Each day, independently, a two-state chain follows rule $M_1 = \begin{pmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{pmatrix}$ with probability $1/2$ or rule $M_2 = \begin{pmatrix} 0.4 & 0.7 \\ 0.6 & 0.3 \end{pmatrix}$ with probability $1/2$ (each $M_{ij}$ is the probability of moving to state $i$ from state $j$). The EFFECTIVE one-step matrix is the average $\bar M = \tfrac12 M_1 + \tfrac12 M_2$. What is the steady-state probability of state $1$ under $\bar M$? Express your answer as a common fraction.