Fair Decision from a 3/5 Coin

A coin lands tails with probability $3/5$ (heads with $2/5$). You want a fair 50-50 decision, so you use this procedure: toss the coin twice. If the two tosses differ, the pair HT means you win and TH means your friend wins. If they are the same (HH or TT), discard both tosses and repeat.

What is the expected total number of coin tosses until a decision is reached? Give an exact fraction.