
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
Find all natural numbers \( n \) for which it is possible to mark \( n \) cells on an \( n \times n \) checkerboard, placing them in different rows and different columns, such that a knight can sequentially visit these \( n \) cells starting from one of them, without stepping on any cell more than once, and return to the initial cell. The knight can only step onto the marked cells.

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