
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
For an integer \( n (n \geqslant 2) \), determine the maximum real number \( C_{n} \) such that for any positive real numbers \( a_{1}, a_{2}, \cdots, a_{n} \), the following inequality holds:
$$
\frac{1}{n} \sum_{i=1}^{n} a_{i}^{2} \geqslant\left(\frac{1}{n} \sum_{i=1}^{n} a_{i}\right)^{2}+C_{n}\left(a_{1}-a_{n}\right)^{2}
$$

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