
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
Find, with proof, the largest possible value of

$$
\frac{x_{1}^{2}+\cdots+x_{n}^{2}}{n}
$$

where real numbers \(x_{1}, \ldots, x_{n} \geq -1\) are satisfying \(x_{1}^{3}+\cdots+x_{n}^{3}=0\).

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