
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
Write either  $1$  or  $-1$  in each of the cells of a  $(2n) \times (2n)$ -table, in such a way that there are exactly  $2n^2$  entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be  $M$ . Determine the largest possible value of  $M$ .

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