
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, and satisfy the equation $a\sin B = \sqrt{3}b\cos A$.  
$(1)$ Find the measure of angle $A$.  
$(2)$ Choose one set of conditions from the following three sets to ensure the existence and uniqueness of $\triangle ABC$, and find the area of $\triangle ABC$.  
Set 1: $a = \sqrt{19}$, $c = 5$;  
Set 2: The altitude $h$ on side $AB$ is $\sqrt{3}$, $a = 3$;  
Set 3: $\cos C = \frac{1}{3}$, $c = 4\sqrt{2}$.

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