F = $\langle y,z,x\rangle$. Surface $S$ is the part of the plane $x+y+z=0$, which can be written as the graph $z=-x-y$ over the disk $D=\{(x,y):x^2+y^2\le1\}$. Choose the upward-pointing normal (positive $z$-component), so the surface parameterization is
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