
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 polynomial \[P(z) = z^4 + 2z^3 + 6z^2 + 8z + 16\] having roots $w_1, w_2, w_3, w_4$. If $g(z) = -2\overline{z}$ transforms each root of $P(z)$ to generate a new polynomial \[Q(z) = z^4 + A'z^3 + B'z^2 + C'z + D'\] where $A', B', C', D'$ are complex numbers. Find the sum $B' + D'$.
A) $40$
B) $104$
C) $280$
D) $312$

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