
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 positive integers \( n \) for which there exists a set \( S \) with the following properties:

(i) \( S \) consists of \( n \) positive integers, all smaller than \( 2^{n-1} \);

(ii) for any two distinct subsets \( A \) and \( B \) of \( S \), the sum of the elements of \( A \) is different from the sum of the elements of \( B \).

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