Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are at a distance of $2$ apart such that $AC < AD$. We can write $\tan \angle BOD = \frac{a}{b}$, where $a$ and $b$ are positive integers such that $\gcd(a, b) = 1$. Compute $a + b$.

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