
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
Given the vertices of the tetrahedron $A(2,3,1)$, $B(4,1,-2)$, $C(6,3,7)$, and $D(-5,-4,8)$, find:

1) The length of the edge $A B$.
2) The angle between the edges $A B$ and $A D$.
3) The angle between the edge $A D$ and the plane $A B C$.
4) The volume of the tetrahedron $A B C D$.
5) The equation of the edge $A B$.
6) The equation of the plane $A B C$.
7) The equation of the height dropped from $D$ onto $A B C$.
8) The projection $O$ of point $D$ onto the base $A B C$.
9) The height $D O$.

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