
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 lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $A=\frac{\pi}{3}$ and $a=3$. <br/>$(1)$ If $\sin B + \sin C = 2\sqrt{3}\sin B \cdot \sin C$, find the value of $\frac{1}{b} + \frac{1}{c}$; <br/>$(2)$ If $b=\sqrt{6}$, let $D$ be a point on the extension of $CA$ such that $BD\perp BC$, find the length of segment $AD$.

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