
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 $\triangle ABC$, $b=2\sqrt{3}$, and from condition ① $\sqrt{3}\cos B = b\sin C$; condition ② $2a-c=2b\cos C$. Choose one of the two conditions as known to answer the following questions.  
$(1)$ If $a=2$, find the area of $\triangle ABC$;  
$(2)$ If $\triangle ABC$ is an acute triangle, find the range of values for $a+c$.  
Note: If you choose to answer using condition ① and condition ② separately, credit will be given for the first answer.

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