
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]


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Given that the eccentricities of a confocal ellipse and a hyperbola are \( e_1 \) and \( e_2 \), respectively, and the length of the minor axis of the ellipse is twice the length of the imaginary axis of the hyperbola, find the maximum value of \( \frac{1}{e_1} + \frac{1}{e_2} \).

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