
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 $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It satisfies the condition $(2a+c)\vec{BC}\cdot\vec{BA}+c\vec{CA}\cdot\vec{CB}=0$.

$(1)$ Calculate the measure of angle $B$.

$(2)$ If $b=2\sqrt{3}$, find the minimum value of $\vec{AB}\cdot\vec{CB}$.

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