
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 \(A_{1}\), \(A_{2}\), \(A_{3}\), and \(A_{4}\), as well as its height dropped from vertex \(A_{4}\) onto the face \(A_{1}A_{2}A_{3}\).

\(A_{1}(2 ;-1 ; 2)\)

\(A_{2}(1 ; 2 ;-1)\)

\(A_{3}(3 ; 2 ; 1)\)

\(A_{4}(-4 ; 2 ; 5)\)

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