
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 the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $sinB=\frac{1}{3}$. Given either condition ① $a^{2}-b^{2}+c^{2}=2$ or condition ② $\overrightarrow{AB}•\overrightarrow{BC}=-1$, fill in the blank above and answer the following questions:
$(Ⅰ)$ Find $ac$;
$(Ⅱ)$ If $sinAsinC=\frac{\sqrt{2}}{3}$, find $b$.

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