$2.$ Alice and Bob are playing a game. There is a linked list with n nodes, each with a single integer li $.$ They take turns choosing either the integer in the head node or tail node $($and removing that node from the list$)$ until the list is empty. The final score for a player is the sum of the numbers chosen by that player. $($a$)$ Give an example where the optimal strategy is not to always take the largest number. Explain your answer. $($b$)$ Design an efficient iterative dynamic programming algorithm to determine the final scores of Alice and Bob, and analyze the runtime. Alice and Bob have a lot of problemsolving skills, due to how many CSCI $170$ examples they appeared in$.$ Therefore, assume that Alice and Bob both play optimally.