Surface Integrals, Stokes & Divergence · Challenge problem

Let $S$ be the upper hemisphere $x^2 + y^2 + z^2 = 1$, $z \ge 0$, with upward normal, and let $\mathbf F = (xz^2,\ x^2y - z^3,\ 2xy + y^2z)$. Find $\iint_S \mathbf F\cdot d\mathbf S$. Give an exact answer in terms of $\pi$.