
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
Describe all possible ways to paint each natural number in one of three colors so that the following condition is met: if the numbers \(a\), \(b\), and \(c\) (not necessarily distinct) satisfy the condition \(2000(a+b) = c\), then they are either all of the same color or each of a different color.

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