Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three vectors such that $\vec{a} \perp (\vec{b} + \vec{c})$, $\vec{b} \perp (\vec{c} + \vec{a})$, and $\vec{c} \perp (\vec{a} + \vec{b})$. If $|\vec{a}| = 1$, $|\vec{b}| = 2$, and $|\vec{c}| = 3$, then $|\vec{a} + \vec{b} + \vec{c}|$ is: