Let $\vec{a}=2 \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+\hat{k}$ be two vectors. Consider a vector $\vec{c}=\alpha \vec{a}+\beta \vec{b}, \alpha, \beta \in \mathbb{R}$. If the projection of $\vec{c}$ on the vector $(\vec{a}+\vec{b})$ is $3 \sqrt{2}$, then the minimum value of $(\vec{c}-(\vec{a} \times \vec{b})) \cdot \vec{c}$ equals