
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
Find the value of $A$, $B$, and $C$, if the cubic equations $x^3 + Ax^2 + 20x + C = 0$ and $x^3 + Bx^2 + 100 = 0$ have two roots in common. Additionally, determine $a + b + c$ where the product of these common roots is expressed as $a \sqrt[b]{c}$, and $a$, $b$, and $c$ are positive integers when simplified.

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