
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
Given the quadratic equation $x^2 + 2mx + 4n = 0$, let $r_1$ and $r_2$ be its roots. Another quadratic equation has roots $r_1^4$ and $r_2^4$. Denote the sum of these roots as $-p$ and their product as $q$. Establish the expressions for $p$ and $q$ in terms of $m$ and $n$.
A) $p = 16m^4 - 64m^2n + 32n^2, q = 256n^4$
B) $p = 8m^4 - 32m^2n + 16n^2, q = 64n^4$
C) $p = 16m^4 - 32m^2n + 64n^2, q = 256n^4$
D) $p = 4m^4 - 16m^2n + 8n^2, q = 128n^4$

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