INMO 2024 Problem 6

For each positive integer $n \geq 3$, define $A_{n}$ and $B_{n}$ as

$$\begin{gathered} A_{n}=\sqrt{n^{2}+1}+\sqrt{n^{2}+3}+\cdots+\sqrt{n^{2}+2 n-1} \\ B_{n}=\sqrt{n^{2}+2}+\sqrt{n^{2}+4}+\cdots+\sqrt{n^{2}+2 n} \end{gathered}$$

Determine all positive integers $n \geq 3$ for which $\left\lfloor A_{n}\right\rfloor=\left\lfloor B_{n}\right\rfloor$.

Note. For any real number $x,\lfloor x\rfloor$ denotes the largest integer $N$ such that $N \leq x$.