
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
In a white grid of size \(2016 \times 2016\), some cells are colored black. A natural number \(k\) is called "lucky" if \(k \leq 2016\) and each \(k \times k\) subgrid within the grid has exactly \(k\) black cells. For example, if all cells are black, then the only lucky number is 1. What is the maximum number of lucky numbers possible?

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