
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. We partition the numbers $1, 2, \ldots, n^{2}$ 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 sum 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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