
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 $n \geq 2$, determine the largest constant $C_{n}$ such that for any real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying $a_{1} + a_{2} + \cdots + a_{n} = 0$, the following inequality holds:
\[
\sum_{i=1}^{n} \min \{a_{i}, a_{i+1}\} \leq C_{n} \cdot \min_{1 \leq i \leq n} a_{i},
\]
where it is understood that $a_{n+1}=a_{1}$.

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