
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
Natural numbers $1, 2, 3, \cdots, n$ are written on a blackboard where $n \geq 3$. Each time, you are allowed to erase any two numbers $p$ and $q$ and replace them with $p + q$ and $|p - q|$. After a finite number of such operations, all the numbers on the blackboard become $k$. What are the possible values of $k$?

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