
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 function $f(x)=A\sin(\omega x+\varphi)$ where $(A > 0, \omega > 0, 0 < \varphi < \frac{π}{2})$, the distance between two adjacent axes of symmetry of the graph of $f(x)$ is $\frac{π}{2}$. The maximum value of the function $f(x)$ is $2$, and _____. Choose one of the following three conditions to fill in the blank: ① $f(x-\frac{π}{6})$ is an odd function; ② $f(0)=\sqrt{3}$; ③ $x=\frac{π}{12}$ is an axis of symmetry of the function $f(x)$. Answer the following questions: <br/>$(1)$ Find the analytical expression of the function $f(x)$; <br/>$(2)$ In triangle $ABC$, $a$, $b$, and $c$ are the side lengths opposite to angles $A$, $B$, and $C$, respectively. If $f(A)=\sqrt{3}$, $c=3$, and the area of $\triangle ABC$ is ${S}_{△ABC}=3\sqrt{3}$, find the value of $a$.

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