Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

The symplectic Euler method for $x' = y$, $y' = -x$ updates $x_{n+1} = x_n + h\,y_n$, then $y_{n+1} = y_n - h\,x_{n+1}$. Run it with $h = \dfrac{1}{2}$ from $(x_0, y_0) = (1, 0)$ forever. Find the supremum of the computed values $x_n$ over all $n \ge 0$.