Let $\vec{a}=\hat{i}+5 \hat{j}+\alpha \hat{k}, \vec{b}=\hat{i}+3 \hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2 \hat{j}+3 \hat{k}$ be three vectors such that, $|\vec{b} \times \vec{c}|=5 \sqrt{3}$ and $\vec{a}$ is perpendicular to $\overrightarrow{\mathrm{b}}$. Then the greatest amongst the values of $|\overrightarrow{\mathrm{a}}|^2$ is $\_\_\_\_$ .