
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  $S$  be a set of positive integers, such that  $n \in S$  if and only if   $$ \sum_{d|n,d<n,d \in S} d \le n $$  Find all positive integers  $n=2^k \cdot p$  where  $k$  is a non-negative integer and  $p$  is an odd prime, such that   $$ \sum_{d|n,d<n,d \in S} d = n $$  

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