
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
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given that $A = \frac{\pi}{3}$ and $\frac{b+c}{\sin B + \sin C} = 2$. Find:<br/>
$(1)$ the length of side $a$;<br/>
$(2)$ if 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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