Evaluate the surface integral ${}_{S}F*dS$ for the given vector field $F$ and the oriented surface $S.$ In other words, find the flux of $F$ across S$.$ For closed surfaces, use the positive $($outward$)$ orientation.

$F(x,y,z)=--yj+z^{3}k,S$ is the part of the cone $z=\sqrt[\phantom{2}]{x^2+y^2}$ between the planes $z=1$ and $z=5$ with downward orientation