Consider the duopoly game as studied in Section 3.6 where player I and II produce nonnegative quantities \(x\) and \(y\) and the unit price on the market is \(12-x-y\). In addition, the players now have a cost of production, which is a cost of 1 per unit for player I and a cost of 2 per unit for player II. Write down the payoff to the players as a modification of (3.6), find their best response functions, and determine the equilibrium of this game. What are the players' profits in equilibrium?