Let $ABCD$ be a regular tetrahedron with side length $12$. Select points $E$, $F$, $G$, $H$ on $AC$, $BC$, $BD$, and $AD$, respectively, such that $AE = BF = BG = AH = 3$. Compute the area of quadrilateral $EFGH$.

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