
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 polynomial \[R(z) = z^4 + 6z^3 + 7z^2 + 8z + 2\] whose roots are $r_1, r_2, r_3, r_4$. Define the transformation of a complex number $z = a + bi$ by the function $g(z) = 3i\overline{z}$. Form the polynomial \[S(z) = z^4 + Pz^3 + Qz^2 + Rz + S\] where $P, Q, R, S$ are complex numbers, and the roots of $S(z)$ are \(g(r_1), g(r_2), g(r_3), g(r_4)\). Find the value of \(Q + S\).
A) -99
B) 63
C) 99
D) 162
E) 225

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