
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
Consider the equation \[(z-2)(z^2+4z+8)(z^2+6z+10)=0.\] Let \( z_k \) for \(1 \leq k \leq 5\) be the complex solutions, which can be written in the form \(x_k + y_ki\). An ellipse \(\mathcal{E}\) passes through the points \((x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y_5)\) in the complex plane. Calculate the eccentricity of \(\mathcal{E}\), expressed as \( \sqrt{\frac{m}{n}} \), where \(m\) and \(n\) are coprime positive integers. Find \(m+n\).
A) 6
B) 8
C) 10
D) 12
E) 14

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