Sum Over Min Die

You have been tasked to find the expected value of a die game in which you are rolling $2$ dice at a time. Your first roll of the dice will all be summed up and will be your starting score. Your next, and final, roll of the $2$ dice will be your divisor, in which you are going to pick the optimal die out of the two to divide your total by, which will be your final score.

$$$$

Calculate the expected value of playing this game optimally.