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If $\vec{a} = \alpha \hat{i} + \beta \hat{j} + 3\hat{k}$, $\vec{b} = -\beta \hat{i} - \alpha \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} - \hat{k}$ such that $\vec{a} \cdot \vec{b} = 1$ and $\vec{b} \cdot \vec{c} = -3$, then $\frac{1}{3}((\vec{a} \times \vec{b}) \cdot \vec{c})$ is equal to _____.
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