Nemotron-IMO-Bench 2026 A12

Find all functions $f: \mathbb{Q}[X]\to \mathbb{Q}[X]$ such that $f(P)+f(Q)=f(P+Q)$ for all $P,Q\in \mathbb{Q}[X]$ and such that $\gcd(P, f(P))=1$ if and only if $P$ is squarefree.