
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]


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Suppose there exist nonzero complex numbers \( a, b, c, \) and \( d \) such that \( k \) is a root of both the equations \( ax^2 + bx + c = 0 \) and \( bx^2 + cx + d = 0 \). Determine all possible values of \( k \), separated by commas.

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