
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 a 9-row by 1000-column table with a total of 9000 numbers, where each of the numbers 1 through 1000 appears exactly 9 times, so that the absolute difference between any two numbers in the same column does not exceed 3. Let \( M \) be the minimum value of the sum of the numbers in any column. Find the maximum value of \( M \).

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