
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
The equations \(x^3 + 2Ax + 20 = 0\) and \(x^3 + Bx^2 + 100 = 0\) share exactly two common roots. Find the product of these common roots expressed in the form \(a \sqrt[b]{c}\), where \(a\), \(b\), and \(c\) are positive integers, and determine the value of \(a + b + c\).

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