ISL 2021 N8

For a polynomial $P(x)$ with integer coefficients let $P^{1}(x)=P(x)$ and $P^{k+1}(x)=$ $P\left(P^{k}(x)\right)$ for $k \geqslant 1$. Find all positive integers $n$ for which there exists a polynomial $P(x)$ with integer coefficients such that for every integer $m \geqslant 1$, the numbers $P^{m}(1), \ldots, P^{m}(n)$ leave exactly $\left\lceil n / 2^{m}\right\rceil$ distinct remainders when divided by $n$.