
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 a $2n \times 2n$ grid, each cell is filled with either the number 1 or -1, such that exactly $2n^2$ cells contain 1 and $2n^2$ cells contain -1. Let $M$ be the minimum value of the sum of absolute values of all row sums and column sums. Determine the maximum possible value of $M$.

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