
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 a polynomial $P(z) = z^4 + 6z^3 + 8z^2 + 12z + 9$. It has four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let $f(z) = -2i\overline{z}$ be a transformation function. Define a new polynomial $Q(z)$ whose roots are $f(z_1)$, $f(z_2)$, $f(z_3)$, and $f(z_4)$, with coefficients $A, B, C, D$. Calculate $B + D$.
A) 64  
B) 144  
C) 176  
D) 192  
E) 256

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