
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
Let \( n>1 \) be a positive integer. Each unit square in an \( n \times n \) grid of squares is colored either black or white, such that the following conditions hold:
- Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge;
- Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge;
- Any \( 2 \times 2 \) subgrid contains at least one square of each color.

Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of \( n \) ).

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