
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
Let  $n$  be a positive integer. A finite set of integers is called  $n$ -divided if there are exactly  $n$  ways to partition this set into two subsets with equal sums. For example, the set  $\{1, 3, 4, 5, 6, 7\}$  is  $2$ -divided because the only ways to partition it into two subsets with equal sums is by dividing it into  $\{1, 3, 4, 5\}$  and  $\{6, 7\}$ , or  $\{1, 5, 7\}$ and  $\{3, 4, 6\}$ . Find all the integers  $n > 0$  for which there exists a  $n$ -divided set.

Proposed by *Martin Rakovsky, France*

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