
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
Fill the numbers \( 1, 2, \cdots, 21^{2} \) into a \( 21 \times 21 \) grid, with each cell containing one number. For each row, calculate the difference between the maximum and minimum numbers, and do the same for each column. Let \( M \) be the maximum value among these 42 differences. Find the smallest possible value of \( M \).

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