
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 the function $f(x)$, defined on the domain $(0, +\infty)$, not be identically zero for $x > 0$, $y \in \mathbb{R}$, and always satisfy $f(x^y) = yf(x)$. If $a > b > c > 1$ and $a$, $b$, $c$ form an arithmetic sequence, determine the relationship between $f(a)f(c)$ and $[f(b)]^2$.

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