
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 the following conditions:  
① $2a\cos C+c=2b$,  
② $\cos ^{2}\frac{{B-C}}{2}-\cos B\cos C=\frac{3}{4}$,  
③ $\left(\sin B+\sin C\right)^{2}=\sin ^{2}A+3\sin B\sin C$,  
Choose one of the three conditions to fill in the blank line below and solve the problem. In triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and ______.  
$(1)$ Find the measure of angle $A$;  
$(2)$ If $a=\sqrt{3}$ and the area of $\triangle ABC$ is $\frac{{\sqrt{3}}}{2}$, find the perimeter of $\triangle ABC$.

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