
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 to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $B=60^{\circ}$ and $b=4$.  
Consider the following statements:  
$(1)$ If $c= \sqrt {3}$, then angle $C$ has two solutions;  
$(2)$ If $\overrightarrow{BC} \cdot \overrightarrow{BA}=12$, then the height from side $AC$ is $3 \sqrt {3}$;  
$(3)$ $a+c$ cannot be $9$.  
Among these statements, the correct ones are \_\_\_\_\_\_.

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