
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 a fixed positive even number. Consider an \( n \times n \) square board, which is divided into \( n^{2} \) unit squares. Two different unit squares on the board are said to be adjacent if they share a common side.

Mark \( N \) unit squares on the board such that every unit square (whether marked or not) is adjacent to at least one marked unit square. Determine the minimum value of \( N \).

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