
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 $\triangle ABC$, it is known that $\sin (A+B)=\sin B+\sin (A-B)$.  
$(1)$ Find $\angle A$;  
$(2)$ If $\overrightarrow{AB} \cdot \overrightarrow{AC}=20$, find the minimum value of $|\overrightarrow{BC}|$.

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