Let $\vec{x}$ be a vector in the plane containing vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. If the vector $\vec{x}$ is perpendicular to $(3 \hat{i}+2 \hat{j}-\hat{k})$ and its projection on $\vec{a}$ is $\frac{17 \sqrt{6}}{2}$, then the value of $|\vec{x}|^2$ is equal to $\_\_\_\_$ .