
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 that the circle \( C: x^{2}+(y-1)^{2}=r^{2} \) and the function \( y=\sin x \) have a unique intersection point, and the x-coordinate of this intersection point is \( \alpha \), find the value of \( \frac{2 \sin 2 \alpha \cos \alpha-4 \cos ^{2} \alpha}{\alpha \cos \alpha} \).

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