
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 complex transformation function $g(z) = 1 - 2i\overline{z}$. Given the polynomial
\[R(z) = z^4 + 6z^3 + 11z^2 + 6z + 1,\]
with roots $z_1$, $z_2$, $z_3$, and $z_4$, find the polynomial
\[S(z) = z^4 + Pz^3 + Qz^2 + Rz + S,\]
whose roots are $g(z_1)$, $g(z_2)$, $g(z_3)$, and $g(z_4)$. What is the sum of $Q + S$?
A) $109 - 100i$
B) $100 + 109i$
C) $100 - 109i$
D) $-109 - 100i$
E) $109 + 100i$

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