
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
The cubic polynomial
\[27x^3 - 12x^2 - 12x - 4 = 0\] has a real root of the form $\frac{\sqrt[3]{d} + \sqrt[3]{e} + 2}{f},$ where $d,$ $e,$ and $f$ are positive integers. Find $d + e + f.$

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