
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, with proof, all functions \( f: \mathbb{R} \setminus \{0\} \rightarrow \mathbb{R} \) such that
\[ 
f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z)) 
\]
for all real \( x, y, z \) such that \( xyz = 1 \).

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