
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 vertex \(A_{4}\) to the face \(A_{1} A_{2} A_{3}\).

Given the coordinates:
\[A_{1}(-3, 4, -7)\]
\[A_{2}(1, 5, -4)\]
\[A_{3}(-5, -2, 0)\]
\[A_{4}(2, 5, 4)\]

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