If $ \vec{a} $, $ \vec{b} $, and $ \vec{c} $ are three non-coplanar vectors and $ \vec{p} $, $ \vec{q} $, and $ \vec{r} $ are vectors defined by $ \vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]} $, $ \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]} $, and $ \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]} $, then the value of $ \left(\frac{\vec{a}}{\vec{a} + \vec{b}}\right) \cdot \vec{p} + \left(\frac{\vec{b}}{\vec{b} + \vec{c}}\right) \cdot \vec{q} + \left(\frac{\vec{c}}{\vec{c} + \vec{a}}\right) \cdot \vec{r} $ is: