
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 \( A_{1} A_{2} A_{3} A_{4} \), let \( S_{i} \) (for \( i = 1, 2, 3, 4 \)) denote the area of the face opposite to vertex \( A_{i} \). Let \( \theta_{ij} \) be the dihedral angle between the edges \( A_{i} A_{j} \). Then:
(I) \( \sum_{1 \leq i < j \leq 4} \cos^{2} \theta_{ij} \geq \frac{2}{3} \);
(II) \( \prod_{1 \leq i < j \leq 4} \cos \theta_{ij} \leq \frac{1}{3^{6}} \).

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