Integer polynomial property

$P$ is a polynomial with integer coefficients. Show that if $a$ is an integer such that

$$P(P(P(a))) = a$$

then

$$P(a) = a$$


Solution Sketch:


This uses the fact that $P(x) - P(y)$ is divisible by $x - y$ to get a cyclic chain of divisibility conditions implying each one in the chain is $\pm1$ times the others. Some assumptions like $P(a) \gt a$ etc lead to contradictions.