Vector Calculus · Challenge problem

Let $C$ be the circle in which the plane $x + y + z = 0$ cuts the sphere $x^2 + y^2 + z^2 = 1$, traversed counterclockwise as seen from the point $(1,1,1)$. Evaluate $\displaystyle\oint_C y\,dx + z\,dy + x\,dz$. Give your answer in terms of $\pi$.