For the control system shown in the block diagram below:

The plant transfer function is given by:

$G(s)=\frac{1}{s^2+6s+5}$

$G_c(s)=K\frac{(s+7)}{s}$

and the controller have the form:

a$.$ Find the Open$-$Loop Transfer Function $($OLTF$).$

b$.$ Sketch $($by hand$)$ the root locus of this system. Show all your work.

c$.$ Determine the range of the gain $K$ required for stability. Also find the values of $$ when the root loci intersect with the imaginary axis $($i$.$e$.$ system is critically stable$).$