
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 the quadrilateral $ABCD$ in the plane, it is known that $\angle BAD=3\angle BCD$, $AB=2$, $AD=\sqrt{2}$, and $BD=\sqrt{10}$.  
$(1)$ If $\angle BDC=\frac{5\pi}{12}$, find $CD$.  
$(2)$ Find the maximum area of triangle $BCD$.

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