Problem $2(26$pt$)$ Consider a system shown below. Answer the following questions and show your process.

$($a$)$ Sketch the root locus. Mark the coordinates of start points, the end points, and the direction of all branches.$(5$ pts$)$

$($b$)$ Find the breakaway and break$-$in points $($if there are any$)$ and mark them on the root locus. $(10$ pts $)$

$($c$)$ Find the number of asymptotes $($if there are any$)$ and mark them on the root locus. $(5$ pts $)$

$($d$)$ According to the Routh$-$Hurwitz criterion, the imaginary axis crossing points locate at $K=1.$ Find the coordinates of the crossing points on the s$-$plane. $(5$ pts $)$

$($e$)$ Looking at the block of the top left corner, $K(s+1),$ what kind of compensator it is$?($P$,$ I, D$,$ PD$,$ PI$,$ ID$,$ or PID$)(1$ pts$).$