12. As shown in the figure, in a square piece of paper $ABCD$ with side length 4, the diagonals $AC$ and $BD$ intersect at $O$. Triangle $AOB$ is cut off, and the remaining part is folded along $OC$ and $OD$ so that $OA$ and $OB$ coincide. Then the volume of the tetrahedron with vertices $A$, $B$, $C$, $D$, and $O$ is $\_\_\_\_$ $\frac { 8 \sqrt { 2 } } { 3 }$. Analysis: The folded solid is a regular triangular pyramid with base edge length 4 and lateral edge length $2 \sqrt { 2 }$, with height $\frac { 2 \sqrt { 6 } } { 3 }$. Therefore, the volume of the tetrahedron is $\frac { 1 } { 3 } \times \frac { 1 } { 2 } \times 16 \times \frac { \sqrt { 3 } } { 2 } \times \frac { 2 \sqrt { 6 } } { 3 } = \frac { 8 \sqrt { 2 } } { 3 }$ [Figure]
(5 points) Given the function $f ( x ) = \left\{ \begin{array} { l } | \lg x | , 0 < x \leqslant 10 \\ - \frac { 1 } { 2 } x + 6 , x > 10 \end{array} \right.$, if $a , b , c$ are mutually distinct and $f ( a ) = f ( b ) = f ( c )$, then the range of $abc$ is
12. As shown in the figure, in a square piece of paper $ABCD$ with side length 4, the diagonals $AC$ and $BD$ intersect at $O$. Triangle $AOB$ is cut off, and the remaining part is folded along $OC$ and $OD$ so that $OA$ and $OB$ coincide. Then the volume of the tetrahedron with vertices $A$, $B$, $C$, $D$, and $O$ is $\_\_\_\_$ $\frac { 8 \sqrt { 2 } } { 3 }$.
Analysis: The folded solid is a regular triangular pyramid with base edge length 4 and lateral edge length $2 \sqrt { 2 }$, with height $\frac { 2 \sqrt { 6 } } { 3 }$. Therefore, the volume of the tetrahedron is $\frac { 1 } { 3 } \times \frac { 1 } { 2 } \times 16 \times \frac { \sqrt { 3 } } { 2 } \times \frac { 2 \sqrt { 6 } } { 3 } = \frac { 8 \sqrt { 2 } } { 3 }$\\
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