Let $\triangle ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 6$. The internal angle bisector of $\angle C$ intersects the side $AB$ at point $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM \parallel AC$ and $DN \parallel BC$. If $(MN)^2 = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, what is the sum of the digits of $|p - q|$?

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