In $\triangle ABC$, $AB = AC = 15$ and $BC = 10$. Points $D$ and $E$ are on sides $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$. Determine the length of $\overline{DE}$ if $\overline{DE}$ divides $\triangle ABC$ into two regions with areas in the ratio 4:1. Let's think step by step and output the final answer within \boxed{}.
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