Surface Integrals, Stokes & Divergence · Superchallenge problem

Let $\mathbf F = (y - z,\ z - x,\ x - y)$ and let $C$ be the great circle in which the plane $x + y + z = 0$ meets the unit sphere $x^2 + y^2 + z^2 = 1$, oriented counterclockwise when viewed from high on the positive $z$-axis. Find $\oint_C \mathbf F\cdot d\mathbf r$. Give an exact answer.