
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 are nonzero complex numbers $p$, $q$, $r$, and $s$ such that $m$ is a root of both equations $px^3 + qx^2 + rx + s = 0$ and $qx^3 + rx^2 + (s+mx)x + p = 0$. Determine all possible values of $m$.

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