
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 that \(\alpha\) and \(\beta\) are real numbers, for any real numbers \(x\), \(y\), and \(z\), it holds that
$$
\alpha(x y + y z + z x) \leqslant M \leqslant \beta\left(x^{2} + y^{2} + z^{2}\right),
$$
where \(M = \sum \sqrt{x^{2} + x y + y^{2}} \sqrt{y^{2} + y z + z^{2}}\,\) and \(\sum\) represents cyclic sums. Find the maximum value of \(\alpha\) and the minimum value of \(\beta\).

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