AUSTRIAN-POLISH 1990 Problem 6

$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$ ) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$ . Show that $a_1 = a_3$ .