
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
Place \(n^2\) dominoes of size \(1 \times 2\) or \(2 \times 1\) on a \(2n \times 2n\) grid without overlapping, such that in every \(2 \times 2\) subgrid there are at least two uncovered cells, and these two cells are in the same row or the same column. Find the number of ways to place the dominoes that satisfy these conditions.

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