
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
In acute triangle $\triangle ABC$, where the internal angles $A$, $B$, and $C$ are opposite to sides $a$, $b$, and $c$ respectively, it is known that $2\sin A\left(c\cos B+b\cos C\right)=\sqrt{3}a$.  
$(1)$ Find the value of angle $A$.  
$(2)$ Given that $c=2\sqrt{2}$ and $a=3$.  
$($i$)$ Find $\sin \left(2C+A\right)$.  
$($ii$)$ Find the area of triangle $\triangle ABC$.

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