
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 $C\_1$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\ (a > 0,b > 0)$ and the hyperbola $C\_2$: $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1\ (a > 0,b > 0)$ with asymptote equations $x± \sqrt {3}y=0$, find the product of the eccentricities of $C\_1$ and $C\_2$.

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