
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
Let \( a, b, c, d \) be nonzero complex numbers such that \( k \) is a common root of the equations \( ax^4 + bx^3 + cx^2 + dx + a = 0 \) and \( bx^4 + cx^3 + dx^2 + ax + b = 0 \). Find all possible values of \( k \).

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