Recall that in the knapsack problem we are given a positive integer knapsack capacity W and a set of n items indexed from $1$ to n$,$ where item i is characterized by a positive integer weight w$\_$i and a positive integer value v$\_$i $,$ and our goal is to determine a maximum$-$value subset of the items with total weight at most W$.$

Consider the variant of the knapsack problem in which we are also given an upper bound K on the number of items that we can put in the knapsack. Present a polynomial$-$time algorithm for this variant under the assumption that each item weight is at most n$^2.$ PLEASE FORMULATE ANSWER IN DESCRIPTIVE SENTENCES WITH AN ACCOMPANYING PROOF$--$ NOT CODE.