"Amortized cost analysis:

If we use two queues $$data queue$$ and $$working queue$$ to implement a stack s by the

following way. Suppose that both queues have no size limit$.$

s$.$push$($item$)$

$1.$ enQueue the item into the data queue.

s$.$pop$()$

$1.$ move all the items except the last one from the data queue to the working queue

$2.$ deQueue from the data queue and return

$3.$ now the names of the data queue and the working queue are swapped. That is$,$ working queue $->$ data queue and data queue $->$ working queue

$($a$)$ Suppose that we would like to analyze the amortized costs using the accounting method. Assume there are k items in the stack before the i$-$th op XB$10$ eration. If the i$-$th operation is a pop operation and we assign $2($k $1)$ amortized cost to it$,$ then how much amortized cost for a push operation should be assigned? Justify your answer.

$($b$)$ If the potential method is used, please define a potential function $$ so that the amortized costs for push and pop are constant time and linear time respectively. Assume that there are k items in the stack before the i$-$th operation. Justify your answer."