
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Choose one of the following conditions:①$a+c=2b\sin(C+\frac{\pi}{6})$; ②$(b+c)(\sin B-\sin C)=(a-c)\sin A$; ③$(2a-c)\cos B=b\cos C$. Solve the following problems under the selected condition:<br/>$(1)$ Find angle $B$;<br/>$(2)$ If $a+c=4\sqrt{3}$ and $\overrightarrow{BA} \cdot \overrightarrow{BC}=6$, find the radius of the incircle of $\triangle ABC$.

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