
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
Cells of a  $n*n$  square are filled with positive integers in the way that in the intersection of the  $i-$ th column and  $j-$ th row, the number  $i+j$  is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to  $n$  and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the  $i-$ th column and  $j-$ row is written  $2n+2-i-j$ .

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