Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{d}}=18$. Then $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{d}}|^2$ is equal to