Dice total divided by the better of two dice

A game uses two fair dice. First you throw both and record their sum as your score. Then you throw both again and divide your score by the value of one of these two dice, choosing whichever die you like; the quotient is your final result. Playing to maximise your result, what is its expected value?

$2779/720\approx3.86$. You always divide by the smaller die. The two throws are independent, so the answer is $7\cdot E[1/\min]$, where $P(\min=k)=\tfrac{13-2k}{36}$. This gives $E[1/\min]=\tfrac{397}{720}$ and the result $\tfrac{2779}{720}$.