
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
Consider the quadratic equation $x^2 + 2mx + 3n = 0$, with roots $r_1$ and $r_2$. Determine the coefficients $p$ and $q$ for a new quadratic equation $x^2 + px + q = 0$ such that its roots are $r_1^3$ and $r_2^3$. Also, find the value of $r_1 + r_2$ under the condition that $m^2 = 3n$.
A) \( p = 8m^3 - 18mn, q = 27n^3 \)
B) \( p = m^3 - 3mn, q = n^3 \)
C) \( p = 4m^3 - 9mn, q = 9n^3 \)
D) \( p = 2m^3 - 6mn, q = 3n^3 \)

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