
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 $f(1)$ for certain real numbers $a$, $b$, and $c$, where the polynomial \[g(x) = x^3 + ax^2 + x + 20\] has three distinct roots. Each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 120x + c.\] Additionally, one root of $g(x)$ is $-100$.

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