In the circuit below, the capacitors C$1$C$\_1$C$1$ and C$2$C$\_2$C$2$ are initially uncharged. The battery voltage is V$0$V$\_0$V$0,$ and the resistors are: R$1=2$RR$\_1=2$RR$1=2$R$,$ R$2=$RR$\_2=$ RR$2=$R$,$ and R$3=3$RR$\_3=3$RR$3=3$R$.$ Switch S$1$S$\_1$S$1$ is closed and then switch S$2$S$\_2$S$2$ is closed.

$($a$)$ What is the battery current immediately after both switches are closed and after both switches have been closed for a very long time, in terms of V$0$V$\_0$V$0$ and RRR$?$

$($b$)$ With both switches closed for a long time, what is the voltage across C$1$C$\_1$C$1$ and the voltage across C$2$C$\_2$C$2,$ in terms of V$0$V$\_0$V$0?$