
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
Given vectors (or complex numbers) \(3x_{1}, x_{2}, \cdots, x_{n}\) with modulus not less than 1, for any vector (or complex number) \(x\), what is the maximum number of vectors \(x_{A}=\sum_{i \in A} x_{i}\) among the \(2^{n}\) possible ones such that the modulus of their difference from \(x\) is less than \(\frac{1}{2}\)?

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