
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
Let \( n \) be a positive even number. Partition the set \(\{1, 2, \cdots, n^2\}\) into two subsets \( A \) and \( B \) such that \( |A| = |B| \) and the sums of the elements in \( A \) and \( B \), denoted \( S_A \) and \( S_B \) respectively, satisfy \(\frac{S_A}{S_B} = \frac{20}{19}\). Determine all such \( n \) for which this partition is possible.

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