
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
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, satisfying $2c+b-2a\cos B=0$.  
$(1)$ Find angle $A$;  
$(2)$ If $a=2\sqrt{3}$, $\overrightarrow{BA}\cdot \overrightarrow{AC}=\frac{3}{2}$, and $AD$ is the median of $\triangle ABC$, find the length of $AD$.

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