$4.)(20$ points$)$ For the same system in Problem $3,$

$($a$)$ find the value of $K$ such that the phase margin is $50$;

$($b$)$ use the phase margin to estimate the damping ratio and predict the percent overshoot

of the closed$-$loop system to a step input;

$($c$)$ use the phase margin to determine the maximum magnitude of the frequency response$\mathrm{.}4.$