$1.($Network Flow$)$ Consider the flow network depicted below, where each directed edge is labeled with its capacity. We are applying the Ford$-$Fulkerson algorithm to determine the maximum flow. The first augmenting path chosen is s$-$a$-$c$-$d$-$t$.$ The second augmenting path chosen is s$-$a$-$b$-$c$-$d$-$t$.$

$($a$)$ Draw the residual networks after each of these two augmentation steps, following the order given above.

$($b$)$ List all possible augmenting paths that could be selected for the third augmentation step.

$($c$)$ Determine the value of the maximum flow in this network.