Cover Time of Simple Random Walk on an n-Cycle

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Markov Chains / Cover Times.

Your Task
A simple random walk starts at vertex 0 on an undirected cycle graph $C_n$ with vertices $\{0, 1, \dots, n-1\}$. At each step, the walk moves to either adjacent vertex with probability $1/2$. What is the expected number of steps until every vertex on the cycle has been visited at least once (the cover time)?

Constraints

• Transitions and increments satisfy the Markov or martingale property with respect to natural filtration.
• Stopping times are almost surely finite.