
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
In the quadratic equation $ax^2 + bx + c = 0$, suppose one root is half of the cube of the other root. What must be the relationship between the coefficients $a$, $b$, and $c$?
A) $b^2 = a^2 + 3ac$
B) $b^2 = a^2(1 + 2\sqrt[3]{4}) + a2^{4/3}$
C) $8b^2 = 27ac$
D) $b^2 - 9ac = 0$

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