
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 triangular prism D-ABC, given that $\angle \mathrm{ACB} = \angle \mathrm{ABD} = 90^{\circ}$, $\mathrm{CA} = \mathrm{CB}$, and $\angle \mathrm{BAD} = 30^{\circ}$, if the projection of point $\mathrm{C}$ onto plane $\mathrm{ABD}$ is precisely on line $\mathrm{AD}$, find the sine of the plane angle of the dihedral angle $\mathrm{C}-\mathrm{AB}-\mathrm{D}$.

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