
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 space, the points \(A(-1, 2, 0), B(5, 2, -1), C(2, -1, 4)\), and \(D(-2, 2, -1)\) are given. Find:

a) the distance from the vertex \(D\) of the tetrahedron \(ABCD\) to the centroid of the base \(ABC\);

b) the equation of the plane \(ABC\);

c) the height of the tetrahedron from the 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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