
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 \( p \), \( q \), and \( r \) be the distinct roots of the polynomial \( x^3 - 20x^2 + 125x - 500 \). It is given that there exist real numbers \( A \), \( B \), and \( C \) such that
\[
\dfrac{1}{s^3 - 20s^2 + 125s - 500} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \dfrac{C}{s-r}
\]
for all \( s \not\in\{p,q,r\} \). What is \( \tfrac1A+\tfrac1B+\tfrac1C \)?
A) 710
B) 720
C) 730
D) 740

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