
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 polynomials \(p(x) = x^3 + Cx^2 - 20 = 0\) and \(q(x) = x^3 + Dx - 80 = 0\) that have two roots in common. The product of these common roots can also be expressed in the form \(a \sqrt[b]{c},\) where \(a,\) \(b,\) and \(c\) are positive integers. Determine \(a + b + c.\)

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