
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 real numbers \( A, B, C \) such that the following inequality \( A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geqslant 0 \) (*) holds for any real numbers \( x, y, z \). What conditions should \( A, B, C \) satisfy? (It is required to write the necessary and sufficient conditions, and limit the expression to only equations or inequalities involving \( A, B, C \)).

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