IMO 2024 Problem 2

Determine all pairs $(a, b)$ of positive integers for which there exist positive integers $g$ and $N$ such that

$$\gcd(a^n + b, b^n + a) = g$$

holds for all integers $n \geqslant N$. (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y$.)