
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 $\alpha$ be a positive real number. Determine the greatest real number \( C \) such that for all positive real numbers \( x, y, \) and \( z \) with \( xy + yz + zx = \alpha \), the inequality

$$
\left(1 + \frac{\alpha}{x^2}\right)\left(1 + \frac{\alpha}{y^2}\right)\left(1 + \frac{\alpha}{z^2}\right) \geq C \left(\frac{x}{z} + \frac{z}{x} + 2\right)
$$

holds. When does equality occur?

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