
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 there exist nonzero complex numbers \( p, q, r, s, \) and \( t \) such that \( k \) is a root of both the equations \( px^4 + qx^3 + rx^2 + sx + t = 0 \) and \( qx^4 + rx^3 + sx^2 + tx + p = 0 \). Determine all possible values of \( k \) under these conditions.

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