
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 \(\operatorname{tg} 2 \alpha=\frac{3}{4}\) with \(\alpha \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), find the values of \(\cos 2 \alpha\) and \(\operatorname{tg} \frac{\alpha}{2}\) when the function \(f(x)=\sin (x+2)+ \sin (\alpha-x)-2 \sin \alpha\) has a minimum value of 0.

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