In $\triangle ABC$, points $D$, $E$, and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$, $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$, and $9 \times DE = EF$, find the side length $BC$.

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