
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 an arbitrary positive real number. Determine for this number  $\alpha$  the greatest real number  $C$  such that the inequality  $$ \left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) $$  is valid for all positive real numbers  $x, y$  and  $z$  satisfying  $xy + yz + zx =\alpha.$   When does equality occur?
*(Proposed by Walther Janous)*

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