
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 positive integer. On a \( 2n \times 2n \) grid, place several dominoes such that for each square in the grid, there is exactly one adjacent square covered by a domino. For each positive integer \( n \), determine the maximum number of dominoes that can be placed according to this requirement.

Note: A domino covers two squares arranged either as \( 1 \times 2 \) or \( 2 \times 1 \). Each domino must cover exactly two squares, and no two dominoes can overlap. Two squares are considered "adjacent" if they share a common edge.

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