
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
Find the positive integers $a, b, c$ such that the real root of the cubic polynomial
\[16x^3 - 4x^2 - 4x - 1 = 0\]
is of the form $\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c}.$

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