
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
Let $D_{1}, D_{2}, \cdots, D_{2n}$ be $2n$ points on the hypotenuse $BC$ of right triangle $\triangle ABC$. Define
$$
\angle D_{i-1} A D_{i} = \alpha_{i} \quad (i=1,2,\cdots,2n+1),
$$
with the condition that $D_{i-1} D_{i} = D_{i} D_{i+1}$ for $i=1,2,\cdots, 2n$, where $D_{0} = B$ and $D_{2n+1} = C$. Determine the value of the expression
$$
\frac{\sin \alpha_{1} \cdot \sin \alpha_{3} \cdots \cdot \sin \alpha_{2n+1}}{\sin \alpha_{2} \cdot \sin \alpha_{4} \cdots \cdot \sin \alpha_{2n}}.
$$

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