
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
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).

Given points:
\( A_{1}(1, -1, 1) \)

\( A_{2}(-2, 0, 3) \)

\( A_{3}(2, 1, -1) \)

\( A_{4}(2, -2, -4) \)

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