
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 $\triangle ABC$, $∠B≠\frac{π}{2}$, $cos2B=\sqrt{3}cosB-1$.  
$(1)$ Find $\angle B$;  
$(2)$ Choose one of the three conditions, condition 1, condition 2, or condition 3, as known to ensure the existence and uniqueness of $\triangle ABC$, and find the area of $\triangle ABC$.  
Condition 1: $sinA=\sqrt{3}sinC$, $b=2$;  
Condition 2: $2b=3a$, $b\sin A=1$;  
Condition 3: $AC=\sqrt{6}$, the height on side $BC$ is $2$.  
Note: If the chosen condition does not meet the requirements, the score for the second question is 0; if multiple suitable conditions are chosen and answered separately, the score will be based on the first answer.

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