
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
Solve for the real root in terms of integers of the cubic equation
\[27x^3 - 24x^2 - 6x - 2 = 0\]
and verify if it can be expressed in the form $\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Then, find $a + b + c$.

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