question_answer

\[\vec{a}\],\[\vec{b}\], and \[\vec{c}\] are vectors such that \[\left| \vec{a}+\vec{b}+3\vec{c} \right|=4\]angle between \[\vec{a}\]and \[\vec{b}\]is \[{{\theta }_{1}}\], between \[\vec{b}\]and \[\vec{c}\] is \[{{\theta }_{2}}\] and between \[\vec{a}\] and \[\vec{c}\] varies \[\left[ \pi /6,2\pi /3 \right].\]Then the maximum value of \[\cos {{\theta }_{1}}+3\cos {{\theta }_{2}}\]is ______.

Answer:

4