8. Let $\triangle A B C$ be a triangle with sidelengths $A B=5, B C=7$, and $C A=6$. Let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $L, M, N$ be the midpoints of sides $B C, C A, A B$, respectively. If the area of the convex hexagon with vertices at $D, E, F, L, M, N$ can be written as $\frac{x \sqrt{y}}{z}$ for positive integers $x, y, z$ with $\operatorname{gcd}(x, z)=1$ and $y$ square-free, find $x+y+z$.

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