
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
Call a nonempty set  $V$  of nonzero integers \emph{victorious} if there exists a polynomial  $P(x)$  with integer coefficients such that  $P(0)=330$  and that  $P(v)=2|v|$  holds for all elements  $v\in V$ . Find the number of victorious sets.

*Proposed by Yannick Yao*

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