If two of the three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are unit vectors such that $\vec{a} + \vec{b} + \vec{c} = 0$ and $2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) + 3 = 0$, then what is the length of the third vector?