Erase and Maximize

You have with you $N$ dice. Each die has $6$ faces, with the faces containing the values $\{1, 2, 3, 4, 5, 6\}$ - one value per face. The orientation of the faces does not matter in this problem. You are also given an integer $S$ ($N \le S \le 6\cdot N$). You must choose $N$ integers $x_1, x_2, \ldots, x_N$ such that: - $1 \le x_i \le 6$ for each $1 \le i \le N$, and - $x_1 + x_2 + \ldots +