
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
For the non-positive real numbers \( x, y, z \) that satisfy the inequality \( k\left(x^{2}-x+1\right)\left(y^{2}-y+1\right)\left(z^{2}-z+1\right) \geqslant (xyz)^{2} - xyz + 1 \), find the minimum value of the real number \( k \).

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