
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 $px^5 + qx^4 + rx^3 + sx^2 + tx + p = 0$ and $qx^5 + rx^4 + sx^3 + tx^2 + px + q = 0.$  Enter all possible values of $k,$ separated by commas.

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