
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 the following conditions:①$4\cos ^{2}A+4\cos B\cos C+1=4\sin B\sin C$；②$sinC-sinB=\frac{{csinC-asinA}}{b}$；③The radius of the circumcircle of triangle $\triangle ABC$ is $2$ and $A \lt B$. Choose one of these three conditions and complete the following question, then solve it.<br/>Given that the internal angles of triangle $\triangle ABC$ are $A$, $B$, and $C$, and the lengths of the opposite sides are $a$, $b$, and $c$, with $a=2\sqrt{3}$,____.<br/>$(1)$ Find the measure of angle $A$;<br/>$(2)$ If $b=3c$, find the area of $\triangle ABC$. Note: If multiple conditions are chosen, solve them separately and give credit to the first solution.

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