
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
Determine all natural numbers  $n$  for which there is a partition of  $\{1,2,...,3n\}$  in  $n$  pairwise disjoint subsets of the form  $\{a,b,c\}$ , such that numbers  $b-a$  and  $c-b$  are different numbers from the set  $\{n-1, n, n+1\}$ .

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