IMC 2025 Problem 10

For any positive integer $N$ , let $S_N$ be the number of pairs of integers $1 \leq a, b \leq N$ such that the number $\left(a^2+a\right)\left(b^2+b\right)$ is a perfect square. Prove that the limit $$\lim _{N \rightarrow \infty} \frac{S_N}{N}$$ exists and find its value.