Up for a Game

Chef is again playing a game with his best friend Garry. As usual, the rules of this game are extremely strange and uncommon. First, they are given a stack of $N$ discs. Each disc has a distinct, non-negative integer written on it. The players exchange turns to make a move. Before the start of the game, they both agree upon a set of positive integers $S$ of size $K$. It is guaranteed that S co