
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
For certain real numbers \(p\), \(q\), and \(r\), the polynomial \[g(x) = x^3 + px^2 + 2x + 20\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + 2x^3 + qx^2 + 200x + r.\] What is \(f(-1)\)?

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