A cyclic quadrilateral $ABCD$ has side lengths $AB = 3$, $BC = AD = 5$, and $CD = 8$. The radius of its circumcircle can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$, $b$, $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$.

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