Diagonalization & Matrix Powers · Superchallenge problem

Let $A = \begin{pmatrix}-4 & 5\\5 & -4\end{pmatrix}$ and let $\mathbf{x}(t)$ solve the second-order system $\mathbf{x}'' = A\mathbf{x}$ with $\mathbf{x}(0) = \begin{pmatrix}1\\0\end{pmatrix}$ and $\mathbf{x}'(0) = \begin{pmatrix}v\\2\end{pmatrix}$. For exactly one real $v$ the solution stays bounded for all $t \ge 0$. Find $v$.