
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. Find the smallest integer \( k \) with the following property: it is possible to mark \( k \) cells on a \( 2n \times 2n \) grid in such a way that there exists a unique partition of the grid into dominoes (either \( 2 \times 1 \) or \( 1 \times 2 \)) such that each domino contains at most one marked cell.

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