
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 triangle $A.\sqrt{21}$, $2\sin 2A\cos A-\sin 3A+\sqrt{3}\cos A=\sqrt{3}$

$(1)$ Find the size of angle $A$;

$(2)$ Given that $a,b,c$ are the sides opposite to angles $A,B,C$ respectively, and $a=1$ and $\sin A+\sin (B-C)=2\sin 2C$, find the area of triangle $A.\sqrt{21}$.

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