
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 $f(x, y, z)$ be a ternary symmetric cubic homogeneous polynomial. Then there exists a unique ordered triplet of real numbers $(a, b, c)$ such that
$$
f(x, y, z) = a g_{1} + b g_{2} + c g_{3},
$$
where $g_{1} = \sum_{\text{cyc}} x(x-y)(x-z)$,
$$
\begin{array}{l}
g_{2} = \sum_{\text{cyc}} (y+z)(x-y)(x-z), \\
g_{3} = x y z,
\end{array}
$$
and for non-negative real numbers $x, y, z$, $f(x, y, z) \geq 0 \Leftrightarrow a, b, c$ are all non-negative real numbers.

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