
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]


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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 \((2000,2024)\).

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