Markov Chains & Long-Run Behavior · Hard problem

A two-state chain has transition matrix $M(p) = \begin{pmatrix} 2p - 0.2 & 0.4 \\ 1.2 - 2p & 0.6 \end{pmatrix}$, where $M_{ij}$ is the probability of moving to state $i$ from state $j$. What is the SMALLEST value of $p$ for which $M(p)$ is a valid transition matrix, and, using that value of $p$, what is the steady-state probability of state $1$? Express your answer as a common fraction.