Let $ABC$ be a triangle with side lengths $13$, $14$, and $15$. The points on the interior of $\triangle ABC$ that are at least $1$ unit away from each side are shaded. The area of the shaded region can be expressed in simplest form as $\frac{m}{n}$. Find the value of $m+n$.

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