When to stop a die game where a six wipes you out

You roll a fair die as many times as you like and keep a running total of the faces. If a $6$ appears, the total resets to zero and the game ends. You may stop before any roll and collect your current total. What stopping rule maximises your expected payout, and what is the expected payout under that rule?

Stop once the total reaches $15$ or more; the expected payout is about $6.15$. With total $n$, one more roll changes the expectation by $\tfrac56\cdot3-\tfrac16n$, which is non-negative only while $n\le15$ (and zero at $15$). Evaluating this threshold rule by backward induction gives roughly $6.154$.