MathNet 1996 — Part 01 Question 4

Let $p(x)$ be the polynomial $x^3 + 14x^2 - 2x + 1$. Let $p^{(n)}(x)$ denote $p(p^{(n-1)}(x))$. Show that there is an integer $N$ such that $p^{(N)}(x) - x$ is divisible by $101$ for all integers $x$.