
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
Each cell of a $100 \times 100$ board is painted either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. For which $n$ can we color the board such that there are exactly $n$ balanced cells? (Cells are considered neighbors if they share a side.)

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