
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 points $A(-1, 2, 0)$, $B(5, 2, -1)$, $C(2, -1, 4)$, and $D(-2, 2, -1)$ in space, find:

a) the distance from vertex $D$ of tetrahedron $ABCD$ to the intersection point of the medians of the base $ABC$;

b) the equation of the plane $ABC$;

c) the height of the tetrahedron from vertex $D$;

d) the angle between lines $BD$ and $AC$;

e) the angle between the faces $ABC$ and $ACD$;

f) the distance between lines $BD$ and $AC$.

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