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Let $\vec{a} = \hat{i} + \alpha \hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} - \alpha \hat{j} + \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8\sqrt{3}$ square units, then $\vec{a} \cdot \vec{b}$ is equal to
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