
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
\[27x^3 - 6x^2 - 6x - 2 = 0\]has a real root of the form \(\frac{\sqrt[3]{p} + \sqrt[3]{q} + 2}{r},\) where \(p,\) \(q,\) and \(r\) are positive integers. Find \(p + q + r.\)

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