
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
Determine the values of $a$, $b$, and $c$ in the real root of the form $\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},$ where $a,$ $b,$ and $c$ are positive integers for the cubic polynomial
\[27x^3 - 6x^2 - 6x - 2 = 0.\]

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