
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. It is known that $a\sin(A+B)=c\sin\frac{B+C}{2}$.  
$(1)$ If $a=4$ and $bc=3$, find the perimeter of $\triangle ABC$.  
$(2)$ Given $b=1$, $c=3$, and a point $D$ on side $BC$ such that $S_{\triangle ABD}=3S_{\triangle ADC}$, find $AD$.

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