
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 $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and $4a\sin B = \sqrt{7}b$.  
$(1)$ If $a = 6$ and $b+c = 8$, find the area of $\triangle ABC$.  
$(2)$ Find the value of $\sin (2A+\frac{2\pi}{3})$.

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