
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 $ABC$, with sides $a$, $b$, and $c$ opposing angles $A$, $B$, and $C$, respectively, it is given that $\cos B \cos C - \sin B \sin C = \frac{1}{2}$ and $a = 2\sqrt{3}$.  
(1) Find the measure of angle $A$.  
(2) If $b = 2$, find the length of side $c$.  
(3) If $b + c = 4$, find the area of $\triangle ABC$.

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