
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 $\triangle ABC$, $b=2\sqrt{3}$. Choose one of the following conditions as known to answer the following questions:<br/>$(1)$ If $a=2$, find the area of $\triangle ABC$;<br/>$(2)$ Find the range of values for $a+c$.<br/>Condition 1: $\sqrt{3}\cos B = b\sin C$; Condition 2: $2a-c=2b\cos C$.<br/>Note: If you choose to answer using Condition 1 or Condition 2 separately, credit will be given for the first answer.

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