
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 table of size $2016 \times 2016$, some cells are colored black. A natural number $k$ is called lucky if $k \leq 2016$, and in every square sub-grid of size $k \times k$ within the table, exactly $k$ cells are colored black. (For example, if all cells are black, then the only lucky number is 1.) What is the maximum number of such lucky numbers that can exist?

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