
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 $\overrightarrow{a}=(\cos \alpha,\sin \alpha)$, $\overrightarrow{b}=(\cos \beta,\sin \beta)$, and $|\overrightarrow{a}-\overrightarrow{b}|=\frac{2\sqrt{5}}{5}$.

$(1)$ Find the value of $\cos(\alpha-\beta)$;  
$(2)$ If $0 < \alpha < \frac{\pi}{2}$, $-\frac{\pi}{2} < \beta < 0$, and $\sin \beta= -\frac{5}{13}$, find $\sin \alpha$.

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