
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 $\overrightarrow{m}=(2,\sin \alpha)$ and $\overrightarrow{n}=(\cos \alpha,-1)$, where $\alpha\in(0, \frac{\pi}{2})$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
$(1)$ Find the values of $\sin 2\alpha$ and $\cos 2\alpha$;
$(2)$ If $\sin (\alpha-\beta)= \frac{\sqrt{10}}{10}$, and $\beta\in(0, \frac{\pi}{2})$, find the angle $\beta$.

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