
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
Recall that the conjugate of the complex number $w = a + bi$ is $\overline{w} = a - bi$. For any complex number $z$, let $g(z) = -2i\hspace{1pt}\overline{z}$. Consider the polynomial \[P(z) = z^4 - 2z^3 + 6z^2 - 3z + 2\] with four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let \[Q(z) = z^4 + Az^3 + Bz^2 + Cz + D\] be the polynomial whose roots are $g(z_1)$, $g(z_2)$, $g(z_3)$, and $g(z_4)$, where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. Determine $B + D$.
- $(\textbf{A}) \: 24$
- $(\textbf{B}) \: 32$
- $(\textbf{C}) \: 48$
- $(\textbf{D}) \: 56$
- $(\textbf{E}) \: 64$

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