
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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

Present the answer in LaTex format: \boxed{Your answer}
user