
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 that all the roots of the polynomial $z^3 + pz^2 + qz + r = 0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ with radius $\frac{1}{2}$, find the sum of the reciprocals of the roots. Which of the following must this sum necessarily be?
- No choices provided yet.
**A)** $-2p$  
**B)** $p$  
**C)** $-q$  
**D)** $-2q$

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