First Return to the Origin on an Asymmetric Cycle

Consider a Markov chain on $\{0, 1, 2, 3\}$ arranged in a cycle. From state $i$, the chain moves clockwise to state $(i+1) \bmod 4$ with probability $p_i$ and counterclockwise to state $(i-1) \bmod 4$ with probability $1 - p_i$, where $$p_0 = \tfrac{3}{4}, \quad p_1 = \tfrac{1}{2}, \quad p_2 = \tfrac{1}{4}, \quad p_3 = \tfrac{1}{2}.$$

Compute the expected number of steps to return to state $0$ for the first time, starting from state $0$.

*Recall:* The mean return time to state $i$ in an irreducible chain equals $1/\pi_i$.