Keep or reroll with a penalty when the total hits seven

Roll a fair die. You may take the shown amount in dollars, or roll a second time. If you roll again and the two rolls total $7$ or more, you must pay out the value of your first roll; if the total is below $7$, you are paid the value of the second roll. With optimal decisions, what is your expected payout?

$67/18$. Rerolling after a first roll of $r$ is worth $\tfrac16\left[\sum_{s\le 6-r}s-r^2\right]$: that is $\tfrac73$ for $r=1$ (beats $1$), but only $1$ for $r=2$ and less for higher $r$. So reroll only a $1$, giving $\tfrac16\left(\tfrac73+2+3+4+5+6\right)=\tfrac{67}{18}$.