$(15\%)$ The NP$-$complete SUBSET$-$SUM problem asks, given a set of non$-$negative integers $S$ and

a target $k,$ whether whether or not a subset from a list of integers can sum to a target value. For

example, consider the list of $S=\{1,2,3,4\}.$ If the target $k=6,$ then there are two subsets that

achieve this sum $\{1,2,3\}$ and $\{2,4\}.$ If $k=12,$ then there are no solutions. The AVERAGE$-$SUM

problem asks, given just a set of non$-$negative integers $S,$ whether or not a subset from a list of

integers can sum to the average of $S($i$.$e$.\frac{1}{|S|}{\displaystyle \underset{\text{i}\text{i}\text{n}\text{S}}{\overset{?}{}}}$i where $|S|$ is the number of elements in $S).$ It

is similar to the SUBSET$-$SUM problem, except now the target value is always the average value

in S$.$ Prove that AVERAGE$-$SUM is NP$-$complete, or give a polynomial$-$time algorithm for it$.$