
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
Suppose real numbers  $A,B,C$  such that for all real numbers  $x,y,z$  the following inequality holds:

\[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\]

Find the necessary and sufficient condition  $A,B,C$  must satisfy (expressed by means of an equality or an inequality).

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