
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
Let the volume of the tetrahedron $ABCD$ be $V$. The areas of the faces opposite the vertices $A, B, C, D$ are $S_{A}, S_{B}, S_{C}, S_{D}$ respectively. Let $BC=a$, $DA=a'$, $CA=b$, $DB=b'$, $AB=c$, $DC=c'$, and the angles between these opposite edges be $\alpha, \beta, \gamma$. Then the following hold:
1. $S_{A}^{2} + S_{B}^{2} + S_{C}^{2} + S_{D}^{2} = \frac{1}{4}\left[(aa' \cdot \sin \alpha)^{2} + (bb' \cdot \sin \beta)^{2} + (cc' \cdot \sin \gamma)^{2}\right]$;
2. $S_{A}^{2} + S_{B}^{2} + S_{C}^{2} + S_{D}^{2} \geq 9(3V^{4})^{\frac{1}{3}}$.

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