
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 $\triangle ABC$, $c=2$, $C=30^{\circ}$. Choose one of the three conditions, Condition 1, Condition 2, or Condition 3, as the given condition to uniquely determine the triangle. Find:<br/>
$(1)$ the value of $a$;<br/>
$(2)$ the area of $\triangle ABC$.<br/>
Condition 1: $2b=\sqrt{3}a$;<br/>
Condition 2: $A=45^{\circ}$;<br/>
Condition 3: $b=2\sqrt{3}$.

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