
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
Given that \( f(x) \) is an \( n \)-th degree polynomial in \( x \) where \( n > 0 \), and for any real number \( x \), it satisfies:
\[ 
8 f\left(x^3\right) - x^6 f(2x) - 2 f\left(x^2\right) + 12 = 0 
\]
find \( f(x) \).

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