
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 Proposition: In the Cartesian coordinate plane $(xOy)$, an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and a triangle $\triangle ABC$ with vertex $B$ on the ellipse, while vertices $A$ and $C$ are the left and right foci of the ellipse, respectively. The eccentricity of the ellipse is $e$. Then, $\frac{\sin A + \sin C}{\sin B} = \frac{1}{e}$. Now, extend this proposition to a hyperbola with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, where $\triangle ABC$ has vertex $B$ on the hyperbola, and vertices $A$ and $C$ are the left and right foci, respectively. The eccentricity of the hyperbola is $e$. Then, we have ___.

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