
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 $p, q, r, s$ be real numbers. Suppose that all the roots of the polynomial $z^4 + pz^3 + qz^2 + rz + s = 0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $2$. What is necessarily the sum of the reciprocals of the roots?
A) $-\frac{p}{2}$
B) $\frac{q}{4}$
C) $\frac{-p}{4}$
D) $\frac{p}{4}$

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