
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
Consider a matrix of size \( n \times n \) whose entries are real numbers of absolute value not exceeding 1, and the sum of all entries is 0. Let \( n \) be an even positive integer. Determine the least number \( C \) such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding \( C \) in absolute value.

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