
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 triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(\sin A + \sin B)(a-b) = c(\sin C - \sqrt{3}\sin B)$.  
$(1)$ Find the measure of angle $A$;  
$(2)$ If $\cos \angle ABC = -\frac{1}{7}$, $D$ is a point on segment $AC$, $\angle ABD = \angle CBD$, $BD = \frac{7\sqrt{7}}{3}$, find $c$.

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