
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 \(x, y, z\) be positive real numbers, and \(k \geq 2\) be a real number. Find the minimum value of
\[
\frac{(x^2 + kx + 1)(y^2 + ky + 1)(z^2 + kz + 1)}{xyz}.
\]

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