Sum of absolute components of a unit null vector

Let a unit vector $v=(v_1 v_2 v_3)^T$ be such that $Av=0$ where

$$A=\begin{pmatrix}\frac56&-\frac13&-\frac16\\-\frac13&\frac13&-\frac13\\-\frac16&-\frac13&\frac56\end{pmatrix}.$$

Then the value of $\sqrt6\,(|v_1|+|v_2|+|v_3|)$ equals __________________