
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 $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, and $a\sin B=\sqrt{3}b\cos A$. Find:<br/>$(Ⅰ)$ the measure of angle $A$;<br/>$(Ⅱ)$ choose one set of conditions from the following three sets to ensure the existence and uniqueness of the triangle, and find the area of $\triangle ABC$. Set 1: $a=\sqrt{19}$, $c=5$; Set 2: $\cos C=\frac{1}{3}$, $c=4\sqrt{2}$; Set 3: the altitude $h$ on side $AB$ is $\sqrt{3}$, $a=3$.

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