Andrew rolls two fair six-sided dice, each numbered from $1$ to $6$. Brian rolls one fair twelve-sided die, numbered from $1$ to $12$. The probability that the sum of the numbers obtained from Andrew's two rolls is less than the number obtained from Brian's roll can be expressed as a common fraction in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let's think step by step and output the final answer within \boxed{}.
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