
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 that an ellipse and a hyperbola centered at the origin share common foci, and their left and right foci are respectively F1 and F2, determine the range of values for the product of the eccentricities of the ellipse and the hyperbola.

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