A polynomial $A(x)$ is said to be simple if $A(x)$ is divisible by $x$ but not divisible by $x^2$. Suppose that a polynomial $P(x)$ has a simple polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists a simple polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{2023}$.