HMMT 2024 T4

Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such a way that the point $(0,0)$ is labeled by 0 , and for every $x, y \geq 0$, the set of numbers labeled on the points $(x, y),(x, y+1)$, and $(x+1, y)$ is $\{n, n+1, n+2\}$ for some nonnegative integer $n$. Determine, with proof, all possible labels for the point $(2000,2024)$.