
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
How many polynomials of the form $x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + 2024$, where $a$, $b$, $c$, $d$, and $e$ are real numbers, have the property that whenever $r$ is a root, both $\frac{-1+i\sqrt{3}}{2} \cdot r$ and $-r$ are also roots?
**A)** 0  
**B)** 1  
**C)** 2  
**D)** 3  
**E)** 4

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