
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
\[27x^3 - 4x^2 - 4x - 1 = 0\] has a real root of the form $\frac{\sqrt[3]{p} + \sqrt[3]{q} + 1}{r},$ where $p,$ $q,$ and $r$ are positive integers. Find $p + q + r.$

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