
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
Find all positive integers  $n$  such that the set  $S=\{1,2,3, \dots 2n\}$  can be divided into  $2$  disjoint subsets  $S_1$  and  $S_2$ , i.e.  $S_1 \cap S_2 = \emptyset$  and  $S_1 \cup S_2 = S$ , such that each one of them has  $n$  elements, and the sum of the elements of  $S_1$  is divisible by the sum of the elements in  $S_2$ .

*Proposed by Viktor Simjanoski*

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