
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 that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a < c$, and $\sin\left(\frac{\pi}{3}-A\right)\cos\left(\frac{\pi}{6}+A\right)=\frac{1}{4}$. Find:

$(1)$ the value of angle $A$;

$(2)$ if $b=\sqrt{3}$, and $a\sin A+c\sin C=4\sqrt{3}\sin B$, find the area of triangle $\triangle ABC$.

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