Let $@$ denote the "averaged with" operation: $a @ b = (a+b)/2$ . Which of the following distributive laws hold for all numbers $x, y,$ and $z$ ? $$\text{I. x @ (y + z) = (x @ y) + (x @ z)}$$ $$\text{II. x + (y @ z) = (x + y) @ (x + z)}$$ $$\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}$$