
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 $m$ 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 $m.$

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