Vector Algebra

If \(\bar{a}, \bar{b}, \bar{c}\) are three vectors such that \(\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}}+\overline{\mathrm{c}})+\overline{\mathrm{b}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})+\overline{\mathrm{c}} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}})=0 \quad\) and \(\quad|\overline{\mathrm{a}}|=1\), \(|\bar{b}|=8\) and \(|\bar{c}|=4\), then \(|\bar{a}+\bar{b}+\bar{c}|\) has the value _________.