
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
A board  $ n \times n$  ( $n \ge 3$ ) is divided into  $n^2$  unit squares. Integers from  $O$  to  $n$  included, are written down: one integer in each unit square, in such a way that the sums of integers in each  $2\times 2$  square of the board are different. Find all  $n$  for which such boards exist.

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