Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

Classical RK4 applied to $y' = \lambda y$ multiplies $y_n$ by $R(z) = 1 + z + \dfrac{z^2}{2} + \dfrac{z^3}{6} + \dfrac{z^4}{24}$, $z = \lambda h$. For a pure oscillation ($\lambda = i$, so $z = ih$ with $h > 0$), find the largest step size $h$ for which $|R(ih)| \le 1$.