$6$A$.$ Let $g(x,y,z)=xyz\sqrt[\phantom{2}]{x^2+y^2+z^2+1}.$ Let vec$(F)=($:$g_x+y,g_y+z,g_z+x$:$).$

$($a$)(5$ points$)$ Calculate curl $(vec(F)).($Hint: You don't need to find $g_x,g_y,g_z).$

$($b$)(5$ points$)$ Let $C$ be intersection of the plane $2x+3y+z=6$ with the cylinder $x^2+y^2=1$ oriented counterclockwise when seen from above. Find the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*vec(d)r.($Hint: Use Stokes' theorem$).$

Show your work, please and thank you.