
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
Let \( a \), \( b \), and \( c \) be distinct roots of the polynomial \( x^3 - 3x^2 - 4x + 12 \). It is given that there exist real numbers \( D \), \( E \), and \( F \) such that:
\[
\frac{1}{s^3 - 3s^2 - 4s + 12} = \frac{D}{s-a} + \frac{E}{s-b} + \frac{F}{s-c}
\]
for all \( s \not\in \{a, b, c\} \). What is \( \frac{1}{D} + \frac{1}{E} + \frac{1}{F} + abc \)?
A) 3
B) 4
C) 5
D) 6

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