
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 $20 \times 20$ grid where each cell is colored black or white, the entire grid is divided into several rectangles with boundaries on the grid lines. For a given coloring pattern $P$, consider all possible divisions in which no rectangle contains more than two black cells. Let $n(P)$ denote the minimum number of rectangles that contain at most one black cell in these divisions. Determine the maximum value of $n(P)$ over all possible coloring patterns $P$.

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