
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
Exactly one integer is written in each square of an  $n$  by  $n$  grid,  $n \ge 3$ . The sum of all of the numbers in any  $2 \times 2$  square is even and the sum of all the numbers in any  $3 \times 3$  square is even. Find all  $n$  for which the sum of all the numbers in the grid is necessarily even.

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