OIMU 2006 Problem 3

Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$ . Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$ .

Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$ .