
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 $4n \times 4n$ (where $n \in \mathbf{N}_{+}$) table of real numbers, each number's absolute value does not exceed 1, and the sum of all numbers is 0. It is known that the absolute value of the sum of the numbers in each row and the absolute value of the sum of the numbers in each column is at least $c$. Find the maximum possible value of $c$.

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