In class, we showed that the cashier's algorithm $($repeatedly taking the largest coin possible$)$ is not optimal for the currency $1,5,7.$ Unfortunately, we have to solve this problem using dynamic programming.

$($a$)$ Let $C(n)$ be the integer$-$valued function that describes the fewest number of coins needed to make change for $n$ cents, using the currency $1,5,7,$ for $n0.$ Describe a mathematical recurrence for $C(n).$

$($b$)$ Use your recurrence to devise a dynamic programming algorithm in pseudocode for computing this function. Your algorithm should be iterative and of the form:

CoINDP$(n)$ :

$\frac{,}{\frac{?}{?}}$ your code here

What's the runtime of your algorithm?