
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 largest real number $C$ such that for all positive real numbers $x, y,$ and $z$ with $x y + y z + z x = \alpha$, the inequality 

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

holds. When does equality hold?

(Walther Janous)

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