
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
Determine the eccentricity of the ellipse passing through the roots of the equation \((z-2)(z^2+1z+2)(z^2+5z+8)=0\) in the complex plane, expressed as \((x_k, y_k)\) where \(1 \leq k \leq 5\). The eccentricity of the ellipse can be expressed in the form \(\sqrt{\frac{m}{n}}\), where \(m\) and \(n\) are coprime integers.
A) 4
B) 5
C) 6
D) 7
E) 8

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