Let $P$ be a plane passing through the points $(1,0,1),(1,-2,1)$ and $(0,1,-2)$. Let a vector $\overrightarrow{\mathbf{a}}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}}$ be such that $\vec{a}$ is parallel to the plane $P$, perpendicular to $(\hat{i}+2 \hat{j}+3 \hat{k})$ and $\vec{a} \cdot(\hat{i}+\hat{j}+2 \hat{k})=2$, then $(\alpha-\beta+\gamma)^2$ equals $\_\_\_\_$ .