
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
Find all triples  $(a,b, c)$  of real numbers for which there exists a non-zero function   $f: R \to R$ , such that   $$ af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0 $$   for all real  $x, y, z$ .

(E. Barabanov)

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