
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 paired in such a way that if the numbers in each pair are added together and the results are multiplied, the product is a square of a natural number.

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