
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 an acute-angled $\triangle ABC$, where $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is known that $\sqrt{3}a=2c\sin {A}$.

$(1)$ Find the size of angle $C$;

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

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