19. (12 points) As shown in the figure, a right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ has volume 4, and the area of $\triangle A _ { 1 } B C$ is $2 \sqrt { 2 }$. (1) Find the distance from $A$ to plane $A _ { 1 } B C$; (2) Let $D$ be the midpoint of $A _ { 1 } C$, with $A A _ { 1 } = A B$ and plane $A _ { 1 } B C \perp$ plane [Figure] $A B B _ { 1 } A _ { 1 }$. Find the sine of the dihedral angle $A - B D - C$.
(12 points)
19. (12 points)\\
As shown in the figure, a right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ has volume 4, and the area of $\triangle A _ { 1 } B C$ is $2 \sqrt { 2 }$.\\
(1) Find the distance from $A$ to plane $A _ { 1 } B C$;\\
(2) Let $D$ be the midpoint of $A _ { 1 } C$, with $A A _ { 1 } = A B$ and plane $A _ { 1 } B C \perp$ plane\\
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$A B B _ { 1 } A _ { 1 }$. Find the sine of the dihedral angle $A - B D - C$.