
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 an even natural number. The numbers \( 1, 2, \ldots, n^2 \) are partitioned into two sets \( A \) and \( B \) of equal size, such that each of the \( n^2 \) numbers belongs to exactly one of the two sets. Let \( S_A \) and \( S_B \) be the sums of all the elements in \( A \) and \( B \) respectively. Determine all \( n \) for which there exists a partition such that

\[ \frac{S_A}{S_B} = \frac{39}{64} \]

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