
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 maximum real number \( k \) such that for any simple graph \( G \) with \( n \) vertices (\( n \geq 3 \)), the inequality \( x^3 \geq k y^2 \) holds, where \( x \) is the number of edges of \( G \) and \( y \) is the number of triangles in \( G \).

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