
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 $\overrightarrow{m}=(\cos \alpha ,-1)$ and $\overrightarrow{n}=(2,\sin \alpha )$, where $\alpha \in \left( 0,\frac{\pi }{2} \right)$, and $\overrightarrow{m}\bot \overrightarrow{n}$.

$(1)$ Find the value of $\cos 2\alpha$;

$(2)$ If $\sin (\alpha -\beta )=\frac{\sqrt{10}}{10}$, and $\beta \in \left( 0,\frac{\pi }{2} \right)$, find the angle $\beta$.

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