Rectangle $ABCD$ has side lengths $AB = 6\sqrt{3}$ and $BC = 8\sqrt{3}$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$, 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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