
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
A *domino* is a  $2 \times 1$  or  $1 \times  2$  tile. Determine in how many ways exactly  $n^2$  dominoes can be placed without overlapping on a  $2n \times  2n$  chessboard so that every  $2 \times  2$  square contains at least two uncovered unit squares which lie in the same row or column.

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