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 \( \left|\vec{a}\times\vec{c}\right|^{2} \) is equal to ______.
