
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 the polynomial $x^7 + ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx - 4040$, where $a$, $b$, $c$, $d$, $e$, and $f$ are real numbers, determine the number of polynomials with the property that if $s$ is a root, then both $\frac{-1+i\sqrt{3}}{2} \cdot s$ and $\frac{-1-i\sqrt{3}}{2} \cdot s$ are also roots.

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