Surface Integrals, Stokes & Divergence · Superchallenge problem

The sphere $x^2 + y^2 + z^2 = 4$ and the cylinder $(x - 1)^2 + y^2 = 1$ meet in a curve; its part with $z \ge 0$ is a closed loop $C$, oriented counterclockwise when viewed from above. For $\mathbf F = (z, x, y)$, find $\oint_C \mathbf F\cdot d\mathbf r$. Give an exact answer.