
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
Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. The areas of the equilateral triangles with side lengths $a$, $b$, $c$ are $S_{1}$, $S_{2}$, $S_{3}$ respectively. Given $S_{1}-S_{2}+S_{3}=\frac{{\sqrt{3}}}{2}$ and $\sin B=\frac{1}{3}$.<br/>$(1)$ Find the area of $\triangle ABC$;<br/>$(2)$ If $\sin A\sin C=\frac{{\sqrt{2}}}{3}$, find $b$.

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