
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 $\overrightarrow{m}=(\cos \alpha ,1-\sin \alpha )$, $\overrightarrow{n}=(-\cos \alpha ,\sin \alpha )$ $(\alpha \in \mathbb{R})$

$(1)$ If $\overrightarrow{m}\perp \overrightarrow{n}$, find the value of angle $\alpha$; 

$(2)$ If $|\overrightarrow{m}-\overrightarrow{n}|=\sqrt{3}$, find the value of $\cos 2\alpha$.

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