
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 equation $(z-2)(z^2+4z+10)(z^2+6z+13)=0$, find the eccentricity of the ellipse passing through the complex plane points corresponding to the solutions in the form $x_k + y_k i$ for $1 \leq k \leq 5$, expressed as $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are coprime positive integers.

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