
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  $f : R - \{0\} \to R$  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}
user