
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
To create a seabed map, the distance between two points $C$ and $D$ on the seabed is measured. A seabed detector at points $A$ and $B$ conducts measurements along the horizontal direction. Points $A$, $B$, $C$, and $D$ lie on the same vertical plane. The detector measures $\angle BAC = 30^\circ$, $\angle DAC = 45^\circ$, $\angle ABD = 45^\circ$, $\angle DBC = 75^\circ$, and the distance between points $A$ and $B$ is $\sqrt{3}$ nautical miles.
(1) Calculate the area of $\triangle ABD$;
(2) Calculate the distance between $C$ and $D$.

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