Surface Integrals, Stokes & Divergence · Superchallenge problem

Let $S$ be the lateral surface of the cylinder $x^2 + y^2 = 1$, $0 \le z \le 1$ (no top or bottom), with outward normal, and let $\mathbf F = (-yz,\ xz,\ 0)$. Use Stokes' theorem, keeping careful track of the two boundary circles, to find $\iint_S (\nabla\times\mathbf F)\cdot d\mathbf S$. Give an exact answer in terms of $\pi$.