$7.($Circulation$)$ Consider the network below, where demand values are shown on vertices $($negative values indicate supply$).$ Lower bounds on flow and edge capacities are represented as $($lower bound, capacity$).$

Your task is to determine if a feasible circulation exists in this graph by completing the following steps:

$($a$)$ Remove the lower bounds on each edge. Write down the new demands on each vertex $\backslash ($ a$,$ b$,$ c$,$ d$,$ e $\backslash )$ in this order.

$($b$)$ Solve the circulation problem without lower bounds. Write down the max$-$flow value.

$($c$)$ Determine whether a feasible circulation exists in the original graph. Explain your answer.