Suppose $F=($:$4y,-6z,2x$:$)$ and $S$ is that portion of the plane $x+2y+3z=1$ that is the first octant of space, oriented counterclockwise as viewed from above.

Find the flux of curl$(F)$ across $S$ and then evaluate the circulation $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr,$ where $C$ is the boundary of $S.$

$($Give an exact answer. Use symbolic notation and fractions where needed.$)$