APMO 2025 Problem 2

Let $\alpha$ and $\beta$ be positive real numbers. Emerald makes a trip in the coordinate plane, starting off from the origin $(0,0)$ . Each minute she moves one unit up or one unit to the right, restricting herself to the region $|x - y|< 2025$ , in the coordinate plane. By the time she visits a point $(x,y)$ she writes down the integer $\lfloor x\alpha +y\beta \rfloor$ on it. It turns out that Emerald wrote each non- negative integer exactly once. Find all the possible pairs $(\alpha ,\beta)$ for which such a trip would be possible.