
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 $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, and their opposite sides are $a$, $b$, and $c$ respectively. If $\cos B\cos C-\sin B\sin C= \frac {1}{2}$,  
$(1)$ find angle $A$;  
$(2)$ if $a=2 \sqrt {3}$ and $b+c=4$, find the area of $\triangle ABC$.

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