$1.$ You are given a sequence of tasks numbered $1,2,3,...,$ n$.$ You get $$ points if you do task i$,$ where $1$ i $$ n$.$ You must do tasks in order $($each next task you do must have higher number than the previous task$),$ but you may choose to skip some tasks. For example, you may do tasks $1,3,4,7($they are in order$)$; however, you can not do tasks $1,5,3,7(5$ and $3$ are in a wrong order$).$ The reason you might choose to skip some tasks is that even though you can do any individual task i and obtain the $$ points, some tasks are so hard that after doing them you will be unable to do any of the immediately following $$ tasks. In other words, if you do task i$,$ then the next task that you could do will be task $($i $++1).($Note that you do not have to do task $($i $++1)$ after task i$.$ It could be more beneficial to skip task $($i $++1)$ and proceed with some task after that$).$

Suppose that you are given the$$ and $$ values $(1$ i $$ n$)$ for all the tasks as input. Devise the most efficient dynamic programming algorithm for choosing a set of tasks to do that maximizes your total points.

$($a$)$ Define a suitable subproblem.

$($b$)$ Give a recursive solution to the subproblem.

$($c$)$ Give a pseudocode for a dynamic programming algorithm that solves the problem. $($d$)$ Analyze the running time of your algorithm.