
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 $10 \times 10$ square, all the cells in the upper left $5 \times 5$ square are colored black, while the remaining cells are white. Into how many polygons can this entire square be divided along the cell boundaries such that in each polygon the number of black cells is three times less than the number of white cells? The polygons do not need to be equal in shape or size.

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