
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 $(z-2)(z^2+3z+5)(z^2+5z+8)=0$, calculate the eccentricity of the unique ellipse that passes through all the solutions $(x_k, y_k)$, where $x_k$ and $y_k$ are real and imaginary parts of the roots respectively.

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