
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 $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. It is known that $A=60^{\circ}$, $a=6$. Consider the following statements:  
(1) If $b= \sqrt{3}$, then $B$ has two solutions;  
(2) If $\vec{AB} \cdot \vec{AC}=12$, then the area of $\triangle ABC$ is $6 \sqrt{3}$;  
(3) $b+c$ cannot equal $13$;  
(4) The maximum value of $(\vec{AB}+ \vec{AC})\cdot \vec{BC}$ is $24 \sqrt{3}$.

Fill in the correct statement numbers on the line ______.

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