$4.(15$ points$)$ The following diagram shows a branch$-$and$-$bound tree search for an integer programming maximization problem. Each node has a number in parenthesis representing the search order and a value $\backslash ($ Z $\backslash )$ indicating the optimal LP relaxation value in that node. Assume that only nodes $5$ and $6$ yield integer solutions $($i$.$e$.,$ their optimal LP relaxation solution turned out to be an integer solution$).($a$)$ What is the best upper bound and lower bound that you can derive from this branch and$-$bound tree? $($b$)$ Now consider that we branch on node $2$ and create two new nodes: $8$ and $9.$ Node $9$ is infeasible and node $8$ has $\backslash ($ Z$=56\mathrm{.}5\backslash ).$ What can you say about the optimal solution of the problem? If you cannot conclude that the optimal solution has been found, which node should we explore next, why?