Vector Calculus · Superchallenge problem

Let $\mathbf F = (y + 3z,\; 4x - z,\; 2y)$. Over all circles of radius $1$ in space, each traversed in either direction, what is the largest possible value of the circulation $\displaystyle\oint_C\mathbf F\cdot d\mathbf r$? Give your answer in terms of $\pi$.