Let $S$ be the portion of the paraboloid $z = 9 - x^2 - y^2$ that lies above the plane $z = 5$, oriented upwards. Let $\vec{F} = \langle y e^z, x^2 z, x y \rangle$. Evaluate the flux integral $\iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$.