Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a} \cdot \vec{b}=0$. For some $x, y \in R$, let $\vec{c}=x \vec{a}+y \vec{b}+(\vec{a} \times \vec{b})$. If $|\vec{c}|=2$ and the vector $\overrightarrow{\mathrm{c}}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, then the value of $8 \cos ^2 \alpha$ is $\_\_\_\_$。