
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
In an \( n \times n \) grid (\( n \geq 3 \)), each cell is filled with a fixed integer. It is known that in any \( 3 \times 3 \) subgrid, the sum of all numbers is even, and in any \( 5 \times 5 \) subgrid, the sum of all numbers is also even. Find all values of \( n \) such that the sum of all numbers in the entire \( n \times n \) grid is even.

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