a$)$ Trace through the Ford$-$Fulkerson max flow algorithm on the input instance given in the figure below. Note that in the first iteration when value of the flow is $0,$ the graph G and the residual graph Gy are one in the same. Start by highlighting the first augmenting path in the figure below, labeled Iteration $1.$ For iterations $2$ and later, indicate $($using the boxes$)$ the current$/$new flow f in G and give the value of the low v$($f$).$ Then $($in the provided space labeled G$,)$ draw the residual graph for that flow, and highlight in G$,$ the next augmenting s$-$t path, giving the bottleneck edge capacity b of the augmenting path $($i$.$e$.,$ the amount by which you are augmenting the flow value$).$ For full credit, always greedily choose the greatest$-$possible augmenting path from G$,$ in each iteration. $($Le$.,$ if there are two augmenting paths, one with bottleneck value $5,$ and the other with bottleneck value $8,$ you should choose the latter.$)$ Again, you should start with simply highlighting the first augmenting s $-$ t path in the top figure for Iteration $1$ here, and giving its bottleneck capacity. b$)$ Certify that the flow you illustrated in part $($a$)$ is indeed the maximum flow, by finding a min s $-$t cut with capacity equal to the flow value. Indicate the cut both on the graph as well as by writing down the partition of the nodes.