
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
The five solutions to the equation \[(z+1)(z^2+6z+10)(z^2+8z+18)=0\] may be written in the form \(x_k+y_ki\) for \(1 \leq k \leq 5\), where \(x_k\) and \(y_k\) are real. Let \(\mathcal E\) be the unique ellipse that passes through the points \((x_1,y_1), (x_2,y_2), (x_3,y_3), (x_4,y_4),\) and \((x_5,y_5)\). Determine the eccentricity of \(\mathcal E\), which can be written in the form \(\sqrt{\frac{m}{n}}\) where \(m\) and \(n\) are relatively prime positive integers. What is \(m+n\)?
A) 5
B) 7
C) 9
D) 11
E) 15

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