
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
Please choose one of the two conditions from "$2\sin A\cos B=2\sin C+\sin B$; $cosA+cos\frac{A}{2}=0$." and fill in the blank line below, then answer the following questions. Given that the opposite sides of the interior angles $A$, $B$, $C$ of triangle $\triangle ABC$ are $a$, $b$, $c$, _____.  
$(1)$ Find $A$;  
$(2)$ Let $AD$ be the angle bisector of $\angle A$, $b+c=10$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the length of segment $AD$.

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