
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 positive integer \( n \geq 2 \). On an infinite grid, \( n \) cells are colored red. A rectangular grid array is called "special" if and only if it contains at least two red cells on its diagonals. A single red cell is also called special. A row or a column grid array is also special if its two ends are red cells. Given a configuration of \( n \) red cells, let \( N \) denote the maximum number of red cells that a special grid array can contain. Determine the minimum value of \( N \) over all possible configurations of \( n \) red cells.

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