
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 equation \[(z-2)(z^2+6z+13)(z^2+8z+18)=0\] in the complex plane, determine the eccentricity of the unique ellipse that passes through the solutions of this equation, where the solutions may be written as $x_k+y_ki$ for $1\le k\le 5$, and express the eccentricity in the form $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers.

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