
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
For which integers  $N$  it is possible to write real numbers into the cells of a square of size  $N \times N$  so that among the sums of each pair of adjacent cells there are all integers from  $1$  to  $2(N-1)N$  (each integer once)?

Maxim Didin

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