Let @ denote the averaged with operation: a @ b=\dfrac{a+b}{2}. Which of the following distributive laws hold for all numbers x, y, and z ?

\begin{aligned} \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) \end{aligned}

**Answer Choices**

A. I only

B. II only

C. III only

D. I and III only

E. II and III only