Surface Integrals, Stokes & Divergence · Challenge problem

Let $C$ be the circle in which the plane $x + y + z = 2$ cuts the sphere $x^2 + y^2 + z^2 = 4$, oriented counterclockwise when viewed from high on the positive $z$-axis. For $\mathbf F = (y^2,\ z^2,\ x^2)$, use Stokes' theorem to find $\oint_C \mathbf F\cdot d\mathbf r$. Give an exact answer.