
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
Suppose that there exist nonzero complex numbers \(a, b, c,\) and \(d\) such that \(k\) is a root of both the equations \(a x^{3} + b x^{2} + c x + d = 0\) and \(b x^{3} + c x^{2} + d x + a = 0\). Find all possible values of \(k\) (including complex values).

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