Surface Integrals, Stokes & Divergence · Challenge problem

A field $\mathbf F$ has constant curl $\nabla\times\mathbf F = (0, 0, 2)$. By Stokes' theorem, $\displaystyle\oint\mathbf F\cdot d\mathbf r$ around the unit circle in the $xy$-plane (counterclockwise) equals $k\pi$. Find $k$.