Let $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$, $\vec{b} = -\hat{j} + \hat{k}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - \vec{a}| = 2\sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$, then $|(\vec{a} \times \vec{b}) \times \vec{c}| =$