Let $\vec{a}$, $\vec{b}$, and $\vec{c}$ be three unit vectors such that the angle between $\vec{a}$ and $\vec{b}$ is $\alpha$, the angle between $\vec{b}$ and $\vec{c}$ is $\beta$, and the angle between $\vec{c}$ and $\vec{a}$ is $\gamma$. If $|\vec{a} + \vec{b} + \vec{c}| = 2$, then $\cos \alpha + \cos \beta + \cos \gamma =$