Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt{3}$. We construct four semicircles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ whose diameters are the segments $AB$, $BC$, $CD$, $DA$. It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4$ (indices taken modulo $4$). Compute the square of the area of $X_1X_2X_3X_4$.

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