$(20$ points$)$

In the circuit shown in the figure, $R_1=R_4=10,R_2=R_3=R_5=5.$

$($a$)$ When the switch $K_1$ is closed and $K_2$ is open, find the total resistance of the external circuit. $(8$ points$)$

$($b$)$ When the switch $K_1$ is open and $K_2$ is closed, find the total resistance of the external circuit. $(4$ points$)$

$($c$)$ When $K_1$ and $K_2$ are both closed, find the total resistance of the external circuit. $(8$ points$)$