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

\( A_{1}(1, 5, -7) \)

\( A_{2}(-3, 6, 3) \)

\( A_{3}(-2, 7, 3) \)

\( A_{4}(-4, 8, -12) \)

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