
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
Non-zero numbers are arranged in  $n \times n$  square ( $n>2$ ). Every number is exactly  $k$  times less than the sum of all the other numbers in the same cross (i.e.,  $2n-2$  numbers written in the same row or column with this number). 
Find all possible  $k$ .

*Proposed by D. Rostovsky, A. Khrabrov, S. Berlov*

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