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

\( A_{1}(7, 2, 4) \)  
\( A_{2}(7, -1, -2) \)  
\( A_{3}(3, 3, 1) \)  
\( A_{4}(-4, 2, 1) \)

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