
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
On the blackboard, natural numbers are written: \(1,2,3, \cdots, n\). \(n \geqslant 3\).

Each time it is allowed to erase any two numbers \(p\) and \(q\), and replace them with \(p+q\) and \(|p-q|\). After a series of such rewrites, all the numbers on the blackboard become \(k\).

What values can \(k\) take?
(The 25th All-Soviet Union Mathematical Olympiad, 1991)

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