
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 \(\alpha\) and \(\beta\) be real numbers. If for any real numbers \(x, y, z\), the inequality \(a(xy + yz + zx) \leq M \leq \beta (x^2 + y^2 + z^2)\) always holds, where \(M = \sqrt{x^2 + xy + y^2} \cdot \sqrt{y^2 + yz + z^2} + \sqrt{y^2 + yz + z^2} \cdot \sqrt{z^2 + zx + x^2} + \sqrt{z^2 + zx + x^2} \cdot \sqrt{x^2 + xy + y^2}\), find the maximum value of \(\alpha\) and the minimum value of \(\beta\).

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