Jim has two fair $6$-sided dice, one whose faces are labelled from $1$ to $6$, and the second whose faces are labelled from $3$ to $8$. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is $9$, if $\frac{m}{n}$ is the probability he rolled the same die twice where $m, n$ are relatively prime positive integers, then what is $m + n$?


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