Coppersmith-Tetali-Winkler Identity for Hitting Times in a Reversible Markov Chain I'm trying to prove the Coppersmith-Tetali-Winkler identity for hitting times ina  reversible Markov chain, that is that for any three states $a$, $b$ and $c$ in a reversible chain
$$ E_a(\tau_b) + E_b(\tau_c) + E_c(\tau_a) = E_a(\tau_c) + E_c(\tau_b) + E_b(\tau_a), \tag{$*$} $$
where $\tau_b = \inf\{n\ge0 \mid X_n=b\}$ is the hitting time of $b$.
Since this is a named identity, I thought I'd be able to find help via Google, but I haven't been able to. I have a hint which says "think of a chain starting at its stationary distribution and then going to $a$, and add this quantity to both sides of $(*)$. Try as I might, I can't get anywhere with this.
If someone could give me a slightly stronger hint, then that'd be most appreciated!

See Did's comment for the answer. I am merely posting this as an answer so that I can accept it and remove this question from the unanswered and/or unaccepted pile. (Answer written by OP)
Do not +1 this answer -- But definitely do +1 Did's comment!
