
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 two hyperbolas $C\_1$ and $C\_2$ with foci on the $x$-axis, the asymptotic lines of $C\_1$ and $C\_2$ have equations $y=k\_1x$ and $y=±k\_2x$ respectively. Denote the eccentricity of $C\_1$ as $e\_1$ and the eccentricity of $C\_2$ as $e\_2$. If $k\_1k\_2=1$, find the minimum value of $e\_1e\_2$.

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