Markov Chains & Long-Run Behavior · Hard problem

Two servers run independently, each as a two-state Markov chain (Up/Down). Server 1 stays Up with probability $4/5$ and, if Down, becomes Up with probability $1/2$; it starts Up. Server 2 stays Up with probability $2/3$ and, if Down, becomes Up with probability $1/4$; it starts Down. What is the probability that BOTH servers are Up after $2$ steps? Express your answer as a common fraction.