
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 $2n \times 2n$ chessboard is covered with $2n^{2}$ pieces of $2 \times 1$ dominoes. According to Exercise 2831, for $n=3$ there exists a horizontal or vertical line along which the board can be cut in half without having to cut any domino in half. For which other values of $n$ is this statement true?

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