Surface Integrals, Stokes & Divergence · Challenge problem

Let $S$ be the part of the paraboloid $z = 4 - x^2 - y^2$ that lies above the plane $z = 2x$, oriented with upward normal, and let $\mathbf F = (yz,\ 2x,\ x^2 + y)$. Find $\iint_S (\nabla\times\mathbf F)\cdot d\mathbf S$. Give an exact answer in terms of $\pi$.