
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 $\triangle ABC$ with internal angles $A$, $B$, $C$ and corresponding sides $a$, $b$, $c$, vectors $\overrightarrow{m}=(a,\sqrt{3}b)$, $\overrightarrow{n}=({cos({\frac{π}{2}-B}),cos(π-A)})$, and $\overrightarrow{m}\bot \overrightarrow{n}$.  
$(1)$ Find the value of angle $A$.  
$(2)$ If $c=3$ and the area of $\triangle ABC$ is $\frac{{3\sqrt{3}}}{2}$, find the value of $a$.

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