Assume you have $n$ liters of water and you want to embottle it by using a number of bottles.

You are given a set of bottle sizes, $\{1,2,$dots,$k\},$ where each number in the set indicates the

capacity of a bottle in liters. You may assume there is an infinite number of bottles with each

capacity given. You should bottle all the water and completely fill any bottles you use.

Problem: To find the number of possible ways to bottle the water.

Example: Input: $n=4$ liters of water and $k=3,$ the set is $\{1,2,3\}($the set indicates that there are $3$

different bottle types with $1,2,$ and $3$ liters capacities$)$ There are $4$ possible ways of bottling as seen

below:

Design a DP algorithm to solve the problem.