
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
There are 6 non-negative integers at the six vertices of a regular hexagon. Define the operation as replacing a number with the absolute difference of the numbers at its neighboring vertices. For which positive integers \( n \), if the sum of the numbers at the vertices is \( n \), is it always possible to apply a series of such operations so that eventually all six vertices have the number 0?

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