
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  $n$  be an even positive integer. We say that two different cells of a  $n \times n$  board are **neighboring** if they have a common side. Find the minimal number of cells on the  $n \times n$  board that must be marked so that any cell (marked or not marked) has a marked neighboring cell.

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