
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 ellipse $C_1: \frac{x^2}{m^2} + y^2 = 1$ $(m>1)$ shares the same foci with the hyperbola $C_2: \frac{x^2}{n^2} - y^2 = 1$ $(n>0)$, let $e_1$ and $e_2$ be the eccentricities of $C_1$ and $C_2$ respectively. Find the relationship between $m$ and $n$ and calculate the product of the eccentricities $e_1 e_2$.

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