Markov Chains & Long-Run Behavior · Hard problem

Solving a puzzle is modelled as a Markov chain with states Start, Partway, Success and Fail (the last two absorbing). From Start, the chain moves to Partway with probability $3/5$ and to Fail with probability $2/5$. From Partway, it moves to Success with probability $1/2$ and to Fail with probability $1/2$. If an attempt fails, a fresh, independent attempt is made from Start. What is the expected number of attempts needed for the first success? Express your answer as a common fraction.