please do this solution in a paper and give correct answer only and don't use AI or chatgpt I will upvote you.

Consider the following problem: $3-$MAKESPAN: Given $n$ jobs with integer durations $d1,$dots, $dn$ and an integer $D,$ determine if these jobs can be scheduled on $3$ machines such that the maximum finishing time of any job is $D.$

$($a$)$ Show that $3-$MAKESPAN is NP$-$complete. Now consider the optimization version of the problem. MIN$-3-$MAKESPAN: Given $n$ jobs with duration $d1,$dots, dn$,$ determine a schedule of these $n$ jobs on $3$ machines that minimizes the maximum finishing time of any job. The optimization version of a problem is usually harder than the decision version. The next question asks you to show this formally.

$($b$)$ Show that MIN$-3-$MAKESPAN is NP$-$hard.