
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
Determine the values of $a$, $b$, and $c$ such that the equations $x^3 + Cx - 24 = 0$ and $x^3 + Dx^2 + 96 = 0$ share two roots. Furthermore, find $a + b + c$, where $a \sqrt[b]{c}$ is the product of these common roots in simplified form.

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