Vector Algebra

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 _________.