CMO 2021 Problem 4

A function $f$ from the positive integers to the positive integers is called Canadian if it satisfies

$$\operatorname{gcd}(f(f(x)), f(x+y))=\operatorname{gcd}(x, y)$$

for all pairs of positive integers $x$ and $y$. Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$.