
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 $ABC$, with angles $A$, $B$, and $C$ opposite to sides $a$, $b$, and $c$ respectively, it is given that $2\cos C \cdot (a\cos B + b\cos A) = c$.

$(1)$ Find angle $C$;

$(2)$ If $c = \sqrt{7}$ and the area of $\triangle ABC$ is $\frac{3\sqrt{3}}{2}$, find $a+b$.

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