
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 there exists a set \( S \) satisfying the following conditions:
1. \( S \) consists of \( n \) positive integers, each of which is less than \( 2^{n-1} \);
2. For any two distinct non-empty subsets \( A \) and \( B \) of \( S \), the sum of the elements in set \( A \) is not equal to the sum of the elements in set \( B \).

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