
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 \(c\) such that the real root of the cubic polynomial
\[ 27x^3 - 6x^2 - 6x -2 = 0 \]
can be expressed as \(\frac{\sqrt[3]{a} + \sqrt[3]{b} + 2}{c},\) where \(a,\) \(b,\) and \(c\) are positive integers. Find \(a + b + c.\)

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