
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]


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For even positive integer  $n$  we put all numbers  $1,2,...,n^2$  into the squares of an  $n\times n$  chessboard (each number appears once and only once).
Let  $S_1$  be the sum of the numbers put in the black squares and  $S_2$  be the sum of the numbers put in the white squares. Find all  $n$  such that we can achieve  $\frac{S_1}{S_2}=\frac{39}{64}.$ 

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