
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 the largest positive integer \( n \) such that for all positive real numbers \( \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} \), the following inequality holds:
$$
\sum_{i=1}^{n} \frac{\alpha_{i}^{2} - \alpha_{i} \alpha_{i+1}}{\alpha_{i}^{2} + \alpha_{i+1}^{2}} \geq 0 \quad (\alpha_{n+1} = \alpha_{1}).
$$

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