Unknown Die with Uniform Sum
Alan has two 6-sided dice. One is a standard fair die with faces labeled
$ through $6$. The other die has unknown face values -- they need not be the integers
$-$6$ and may be repeated -- but all values are positive integers.
Whenever Alan rolls both dice together, each integer sum from
$ to
2$ (inclusive) is equally likely. What is the sum of all six face values on the unknown die?
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