Markov Chains & Long-Run Behavior · Challenge problem

Ehrenfest urn model: four balls are split between urns A and B. Each step, one of the four balls is chosen uniformly at random and moved to the other urn. Let the state be the number of balls in urn A. Starting with two balls in each urn, what is the expected number of steps until the urns are again split two and two? Express your answer as a common fraction.