
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 $\triangle ABC$, $2\sin A \cos B + b \sin 2A + 2\sqrt{3} a \cos C = 0$.  
$(1)$ Find the measure of angle $C$.  
$(2)$ If $AC = 2$, $CD = \sqrt{3}$, and point $D$ is the midpoint of $AB$, find the area of triangle $\triangle BCD$.

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