
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 vectors $\overrightarrow{m}=(a-\sin\theta, -\frac{1}{2})$, $\overrightarrow{n}=(\frac{1}{2}, \cos\theta)$.  
$(1)$ When $a=\frac{\sqrt{2}}{2}$ and $\overrightarrow{m} \perp \overrightarrow{n}$, find the value of $\sin 2\theta$;  
$(2)$ When $a=0$ and $\overrightarrow{m} \parallel \overrightarrow{n}$, find the value of $\tan\theta$.

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