$[$Adapted from textbook$]$ A laser beam can be used to weld, drill, etch, cut. and mark metals, as shown In Figure $($a$).$ Assume we have a work requirement for an accurate laser to mark a parabolic path with a closed$-$loop control system, as shown in Figure $($b$).$ Calculate the necessary gain to result in a steady$-$state error of $5$ mm for $r(t)=t^{2}cm.$

$($a$)$