Directed Graphs and Network Flows · Medium problem

A flow network has source $s$, sink $t$, and vertices $a,b,c$, with capacities $c(s,a)=6$, $c(s,b)=4$, $c(a,b)=2$, $c(a,c)=4$, $c(b,c)=3$, $c(b,t)=3$, $c(c,t)=6$. It already carries a flow of value $3$, sending $3$ units along $s\to a\to c\to t$ and nothing else. Using the still-untouched path $s\to b\to c\to t$, push its bottleneck. What is the flow's new total value?