Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Let $D$, $E$, and $F$ be the feet of the altitude, angle bisector, and median from $A$ to $BC$, respectively. If $DE = 3$ and $EF = 5$, compute the length of $BC$.

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