
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 \( n \) that satisfy the condition: the numbers \( 1, 2, 3, ..., 2n \) can be divided into pairs such that if you sum the numbers in each pair and multiply the results, the product is a square of a natural number.

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