Surface Integrals, Stokes & Divergence · Superchallenge problem

Let $\mathbf F = \left(\dfrac{x}{x^2 + y^2},\ \dfrac{y}{x^2 + y^2},\ 0\right)$ (defined off the $z$-axis), and let $S$ be the sphere $(x - 1)^2 + y^2 + (z - 2)^2 = 4$ with outward normal. Find $\iint_S \mathbf F\cdot d\mathbf S$. Give an exact answer.