
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
In the tetrahedron \( A_{1}A_{2}A_{3}A_{4} \), the lines corresponding to the midpoints of opposite edges are \( m_{1}, m_{2}, m_{3} \), the centroids of the faces opposite to vertex \( A_{i} \) are \( G_{i} \), and the volume of the tetrahedron is denoted by \( V \). Then:

(I) \( m_{1} \cdot m_{2} \cdot m_{3} \geqslant 3V \);

(II) \( \sum_{1 \leqslant i < j \leqslant 4} A_{i}A_{j} - \frac{27}{16} \sum_{i=1}^{4} A_{i}G_{i}^{2} \geqslant 3 \sqrt[3]{9V^{2}} \);

(III) \( \sum_{i=1}^{4} A_{i}G_{i}^{2} \geqslant \frac{16}{3} \sqrt[3]{9V^{2}} \).

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