
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 all real numbers \(a\), \(b\), and \(c\) such that the polynomial \(x^3 + ax^2 + bx + c\) has three real roots \(\alpha, \beta, \gamma\) (not necessarily all distinct) and the equation \(x^3 + a^3 x^2 + b^3 x + c^3\) has roots \(\alpha^3, \beta^3, \gamma^3\).

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