
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
Given that in $\triangle ABC$, the internal angles $A$, $B$, $C$ satisfy $\sin 2A+\sin (A-B+C)=\sin (C-A-B)+ \frac {1}{2}$, and the area $S$ satisfies $1\leqslant S\leqslant 2$. Let $a$, $b$, $c$ respectively be the sides opposite to angles $A$, $B$, $C$. Consider the following statements:  
$(1) bc(b+c) > 8$  
$(2) ab(a+b) > 16 \sqrt {2}$  
$(3) 6\leqslant abc\leqslant 12$  
$(4) 12\leqslant abc\leqslant 24$  
Among these, the incorrect ones are __________ (Fill in all applicable numbers).

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