
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 an integer \( n \geq 2 \), find the smallest positive real number \( c \) such that for any complex numbers \( z_1, z_2, \ldots, z_n \), the following inequality holds:
$$
\left|\sum_{i=1}^{n} z_{i}\right| + c \sum_{1 \leq i < j \leq n} \left|z_{i} - z_{j}\right| \geq \sum_{i=1}^{n} \left|z_{i}\right|.
$$

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