Markov Chains & Long-Run Behavior · Hard problem

A support duty rotates among $4$ engineers arranged in a circle. Each day, with probability $1/2$ the duty stays with the same engineer, and with probability $1/4$ each it passes to the engineer immediately clockwise or counterclockwise. First check that this transition matrix is doubly stochastic (both its rows and its columns sum to $1$), then use that fact to find the expected number of days until the duty returns to the engineer who had it today.