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