$5.$ In year $2264$ the twenty$-$third starship came o$$ the assembly lines at NASA. This starship was called

the USS Enterprise. Unfortunately, the core libraries of the Enterprise were corrupted during an

exploration mission. The only uncorrupted data structure left was a simple stack. A team of engineers

set out to reimplement all other data structures in terms of stacks, and they started out with queues.

$($a$)(5$ points$)$ The following are parts of their original implementation of queue using two stacks

$($in stack and out stack$).$ Analyze the worst$-$case running times of its enqueue and dequeue

methods, and express them using $$Big$-$Oh$$ notation.