
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 nonzero complex numbers \(a, b, c, d,\) and \(e\) satisfy that \(k\) is a root of both equations \(ax^3 + bx^2 + cx + d = e\) and \(bx^3 + cx^2 + dx + e = a\). Enter all possible values of \(k\).

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