
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
Choose one of the three conditions, ① $\frac{b}{a}=\frac{{cosB+1}}{{\sqrt{3}sinA}}$, ② $2b\sin A=a\tan B$, ③ $c-a=b\cos A-a\cos B$, as the known condition, supplement it in the question below, and then answer the complete question. In triangle $\triangle ABC$, where the internal angles $A$, $B$, and $C$ are opposite sides $a$, $b$, and $c$, respectively, it is known as _____ (just fill in the number).  
$(1)$ Find angle $B$;  
$(2)$ If $\triangle ABC$ is an acute triangle with side length $c=2$, find the range of values for the area of $\triangle ABC$.  
Note: If multiple conditions are selected and answered separately, the scoring will be based on the first answer.

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