
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 \( n \geq 3 \). Find the largest real number \( M \) such that for any sequence of positive real numbers \( x_{1}, x_{2}, \cdots, x_{n} \), there exists a permutation \( y_{1}, y_{2}, \cdots, y_{n} \) satisfying
\[ 
\sum_{i=1}^{n} \frac{y_{i}^{2}}{y_{i+1}^{2} - y_{i+1} y_{i+2} + y_{i+2}^{2}} \geq M,
\]
where \( y_{n+1} = y_{1} \) and \( y_{n+2} = y_{2} \).

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