
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
In the Cartesian coordinate system $xOy$, let vector $\overrightarrow{a}=(\cos \alpha,\sin \alpha)$, $\overrightarrow{b}=(-\sin \beta,\cos \beta)$, and $\overrightarrow{c}=(- \frac {1}{2}, \frac { \sqrt {3}}{2})$.  
$(1)$ If $| \overrightarrow{a}+ \overrightarrow{b}|=| \overrightarrow{c}|$, find the value of $\sin (\alpha-\beta)$;  
$(2)$ Let $\alpha= \frac {5\pi}{6}$, $0 < \beta < \pi$, and $\overrightarrow{a}//( \overrightarrow{b}+ \overrightarrow{c})$, find the value of $\beta$.

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