
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 that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, respectively, and $2\sin \frac {7\pi}{6}\sin \left( \frac {\pi}{6}+C\right)+\cos C= - \frac {1}{2}$.  
$(1)$ Find $C$;  
$(2)$ If $c= \sqrt {13}$ and the area of $\triangle ABC$ is $3 \sqrt {3}$, find the value of $\sin A+\sin B$.

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