Surface Integrals, Stokes & Divergence · Challenge problem

Let $\mathbf F = (y, z, x)$ and let $C$ be the ellipse in which the plane $x + y + z = 2$ cuts the cylinder $x^2 + y^2 = 1$, oriented counterclockwise when viewed from above. Use Stokes' theorem to find $\oint_C \mathbf F\cdot d\mathbf r$. Give an exact answer in terms of $\pi$.