
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, and the area is $S$. It is known that $6S = b(a+c)$.  
$(1)$ If $\sin A = \frac{2}{3}$, find $\cos B$;  
$(2)$ If $b=3$ and $B=\frac{\pi}{3}$, find the area $S$ of triangle $\triangle ABC$.

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