
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
Given the coordinates of the vertices of a pyramid \( A_{1}(4, 2, 5) \), \( A_{2}(0, 7, 2) \), \( A_{3}(0, 2, 7) \), \( A_{4}(1, 5, 0) \). Find:
1. The equations of the planes \( A_{1} A_{2} A_{3} \) and \( A_{1} A_{2} A_{4} \).
2. The angle between the edge \( A_{1} A_{3} \) and the face \( A_{1} A_{2} A_{4} \).
3. The distance from vertex \( A_{3} \) to the face \( A_{1} A_{2} A_{4} \).
4. The shortest distance between the lines \( A_{1} A_{2} \) and \( A_{3} A_{4} \).
5. The equation of the altitude dropped from vertex \( A_{3} \) to the face \( A_{1} A_{2} A_{4} \).

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