Consider the following program that runs on input $n$ :

Set $S=1,$ and $k=1$

while $kn$

$($a$)$ Replace $S$ with $2S$

$($b$)$ Replace $k$ with $k+1$

endwhile

Print $S$

$($a$)$ How many steps does the program take $($as a function of $n)?$ Explain your reasoning.

$($b$)$ What is the function of $n$ that it calculates? $($Your answer should be something like $"$On input $n,$ the algorithm prints $n^{2n}$ except, of course, $n^2$ is not correct.$)$

i$.$ Does the algorithm work on $n=0$ or does $n$ have to be at least $1?$