BALTIC-WAY 2024 Problem 9

Let $S$ be a finite set. For a positive integer $n$, we say that a function $f: S \rightarrow S$ is an $n$-th power if there exists some function $g: S \rightarrow S$ such that

$$f(x)=\underbrace{g(g(\ldots g(x) \ldots))}_{g \text { applied } n \text { times }}$$

for each $x \in S$. Suppose that a function $f: S \rightarrow S$ is an $n$-th power for each positive integer $n$. Is it necessarily true that $f(f(x))=f(x)$ for each $x \in S$ ?