KöMaL 2019 B5012

Let $f(x)$ be a polynomial with integer coefficients. Let $f^{(n)}$ denote the $n$-fold application of $f$: $f^{(n)}(x)=\underbrace{f\Big(f\big(\ldots f}_{n}(x)\ldots\big)\!\Big).$ Let $k(f)$ denote the smallest positive integer $k$ for which $f^{(k)}(x)\equiv x\pmod{13}$ holds for all integers $x$, provided that there exists such a $k$, and let $k(f)=0$ otherwise. Prove that there is a maximum value of $k(f)$, and determine this maximum value.