Singular Values & the SVD · Hard problem

Let $A = \begin{pmatrix}1 & 2 & 0\\2 & 0 & 2\end{pmatrix}$. The maximum of $\|Ax\|$ over unit vectors $x \in \mathbb R^3$ is attained at $x = \pm v_1$, the first right singular vector. Taking the sign with positive first component, find the sum of the three components of $v_1$. Give your answer in simplest radical form.