Numerical Methods: Euler to Runge–Kutta · Challenge problem

Euler's method is applied to the linear system $\mathbf{x}' = A\mathbf{x}$ where $A$ has eigenvalues $-1 \pm 2i$. The iterates stay bounded for every initial vector exactly when $|1 + h\lambda| \le 1$ for both eigenvalues $\lambda$. What is the largest step size $h$ for which this holds? Express your answer as a common fraction.