
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 in triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.  
$(1)$ If $B=\frac{π}{3}$, $b=\sqrt{7}$, and the area of $\triangle ABC$ is $S=\frac{{3\sqrt{3}}}{2}$, find the value of $a+c$.  
$(2)$ If $2\cos C(\overrightarrow{BA}•\overrightarrow{BC}+\overrightarrow{AB}•\overrightarrow{AC})=c^{2}$, find the value of angle $C$.

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