
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
Compute the number of non-empty subsets  $S$  of  $\{-3, -2, -1, 0, 1, 2, 3\}$  with the following property: for any  $k \ge 1$  distinct elements  $a_1, \dots, a_k \in S$  we have  $a_1 + \dots + a_k \neq 0$ .

*Proposed by Evan Chen*

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