$ABC$ is an isosceles triangle with $AB = AC$ and $BC = 48$. $DEFG$ is a square inscribed in triangle $ABC$ such that vertices $D$ and $E$ lie on $AB$ and $AC$ respectively, while vertices $F$ and $G$ lie on $BC$. If $[ADE] = 24$, what is the side length of the square $DEFG$?

**Details and assumptions**

$[PQRS]$ denotes the area of figure $PQRS$.

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