
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
The numbers from  $1$  to  $n^2$  are written in order in a grid of  $n \times n$ , one number in each square, in such a way that the first row contains the numbers from  $1$  to  $n$  from left to right; the second row contains the numbers  $n + 1$  to  $2n$  from left to right, and so on and so forth. An allowed move on the grid consists in choosing any two adjacent squares (i.e. two squares that share a side), and add (or subtract) the same integer to both of the numbers that appear on those squares. 

Find all values of  $n$  for which it is possible to make every squares to display  $0$  after making any number of moves as necessary and, for those cases in which it is possible, find the minimum number of moves that are necessary to do this. 

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