
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 5x5 grid with twenty-five numbers using five 1's, five 2's, five 3's, five 4's, and five 5's (one number per cell) such that the absolute difference between any two numbers in the same column does not exceed 2. Considering the sums of the numbers in each column, let the minimum value of these sums be $M$. Find the maximum value of $M$.

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