$.$ Design a greedy algorithm for arranging the queuing order in a supermarket. Suppose there are n customers come to the counter at the same time, noted as c$1,$c$2,...,$cn$,$ the time to service i$-$th customer is si$,$ i $=1,2,...,$n$,$ and the absolute time to finish i$-$th customer is Ti$,$ i $=1,2,...,$n$.$ Your goal is to decide a queuing order of n customers to minimize the accumulated completion time $($waiting time $+$ service time$)$ of all n customers, that is$,$ to minimize n i$=1$ Ti$.($a$)$ Provide an algorithm to solve this issue; $($b$)$ Prove the correctness of your algorithm by showing the greedy choice property and optimal sub structure; $($c$)$ Justify the running time of your algorithm.