$(12+10\%)$ A large reservoir at temperature $T_R$ is placed in thermal contact with a small system at temperature $T_S.$ They both end up at the temperature of the reservoir, $T_R.$ The heat transferred from the reservoir to the system is $Q=C(T_R-T_S),$ where $C$ is the heat capacity of the system. $($a$)$ Derive the expressions for the entropy changes for the system $S_S$ and the reservoir $S_R.($b$)$ Show that the entropy change in the Universe $S_U-=S_R+S_S$ is larger than $0$ for both $T_R>T_S$ and $T_R.$