$\overline{a},\overline{b},\overline{c}$  are unit vectors such that $|\overline{a}+\overline{b}+\overline{3c}|=4$  . $\left(\overline{a},\overline{b}\right)={\theta }_1,\left(\overline{b},\overline{c}\right)={\theta }_2$  and angle between  $\overline{a}$  and   $\overline{c}$ is  ${\theta }_3$ and  ${\theta }_3$  varies in  $[\frac{\pi }{6},\frac{2\pi }{3}]$ ,. Then the maximum value of  $\cos {\theta }_1+3\cos {\theta }_2=$ _______