
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
Given a $2021 \times 2021$ square table, each cell is colored in one of $n$ colors. It is known that for any four cells of the same color, located in a single column, there are no cells of the same color to the right of the top cell and to the left of the bottom cell. What is the minimum $n$ for which this is possible?

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