
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
Given real numbers \(a, b, c\), the polynomial 
$$
g(x) = x^{3} + a x^{2} + x + 10
$$ 
has three distinct roots, and these three roots are also roots of the polynomial
$$
f(x) = x^{4} + x^{3} + b x^{2} + 100 x + c.
$$

Evaluate the value of \(f(1)\).

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