Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $$\vec{a} \cdot \vec{c}=11, \vec{b} \cdot(\vec{a} \times \vec{c})=27 and  \vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|, then |\vec{a} \times \vec{c}|^{2}$$ is equal to _________.