Consider a triangle $ABC$ with $AB = 13$, $BC = 14$, $CA = 15$. A line perpendicular to $BC$ divides the interior of $\triangle ABC$ into two regions of equal area. Suppose that the aforesaid perpendicular cuts $BC$ at $D$, and cuts $\triangle ABC$ again at $E$. If $L$ is the length of the line segment $DE$, find $L^2$.

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