
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. Determine the smallest positive integer  $k$  with the following property: it is possible to mark  $k$  cells on a  $2n \times 2n$  board so that there exists a unique partition of the board into  $1 \times 2$  and  $2 \times 1$  dominoes, none of which contain two marked cells.


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