Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the center $O$ lies between these chords. Suppose $AB = 6$ and $CD = 8$. Further, suppose that the area of the part of the circle lying between the chords $AB$ and $CD$ is given by \( \frac{m\pi + n}{k} \), where $m$, $n$, and $k$ are positive integers with \( \gcd(m, n, k) = 1 \). What is the value of $m + n + k$?

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