
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  $n\geq 3$  be an integer. Find the largest real number  $M$  such that for any positive real numbers  $x_1,x_2,\cdots,x_n$ , there exists an arrangement  $y_1,y_2,\cdots,y_n$  of real numbers 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  $y_{n+1}=y_1,y_{n+2}=y_2$ .

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