
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 in right triangle $ABC$, $|AB|=3$, $|AC|=1$, $A=\frac{\pi}{2}$, a hyperbola with foci at $B$ and $C$ passes through point $A$ and intersects side $AB$ at point $D$. The equation of the hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$). Find the value of $\frac{|AD|}{|BD|}$.

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