
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. $S$ is composed of $n$ positive integers, each of which is smaller than $2^{n-1}$;
2. For any two different non-empty subsets $A$ and $B$ of $S$, the sum of the elements in subset $A$ is not equal to the sum of the elements in subset $B$.

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