Fair Dice

Two six-sided dice are rolled and their faces are added. The first die is a standard die with faces $1,2,3,4,5,6$. You may relabel the six faces of the second die with any non-negative integers (repeats allowed) so that every total from $1$ to $12$ is equally likely, each with probability $\frac{1}{12}$.

What is the sum of the six faces of your relabelled second die?