$($a$)$ Trace an algorithm from the Ford$-$Fulkerson family of algorithms on the following flow network; suppose the algorithm picks the augmenting path that is lexicographically smallest at each iteration. To save space, it suffices to specify the selected augmenting path used at each iteration of the algorithm and the amount of flow passed through that path. Also show the final residual network. Describe each path as a sequence of vertices on the path $($e$.$g$.,\backslash ($ s B G t $\backslash )).$

$($b$)$ Show the minimum cut associated with the flow you calculated in part $($a$).$