
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 that \( n \) is a positive integer, find the smallest positive integer \( k \) satisfying the following condition: In a \( 2n \times 2n \) grid, mark \( k \) cells such that there exists a unique way to partition the \( 2n \times 2n \) grid into \( 1 \times 2 \) and \( 2 \times 1 \) dominoes, with each domino not containing two marked cells.

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