
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
Let \( x_{1}, x_{2}, \cdots, x_{n} \) be real numbers that are not less than -1, satisfying \( \sum_{i=1}^{n} x_{i}^{3}=0 \). Find the smallest constant \( c \) such that for all positive integers \( n \), \( \sum_{i=1}^{n} x_{i}^{2} \leqslant c n \) holds.

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