
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
For a finite set  $A$  of positive integers and its subset  $B$ , call  $B$  a *half subset* of  $A$  when it satisfies the equation  $\sum_{a\in A}a=2\sum_{b\in B}b$ . For example, if  $A=\{1,2,3\}$ , then  $\{1,2\}$  and  $\{3\}$  are half subset of  $A$ . Determine all positive integers  $n$  such that there exists a finite set  $A$  which has exactly  $n$  half subsets.

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