
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, each cell is filled with one of the numbers from \( 1 \) to \( n^2 \). If, no matter how the grid is filled, there are always two adjacent cells whose numbers differ by at least 1011, find the smallest possible value of \( n \).

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