
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 pyramid \( SABC \), the product of the edge lengths for each of the four faces is equal to the same number. The length of the height of the pyramid, dropped from \( S \) to the face \( ABC \), is \( 2 \sqrt{\frac{102}{55}} \), and the angle \( CAB \) is \( \arccos \left(\frac{1}{6} \sqrt{\frac{17}{2}}\right) \). Find the volume of the pyramid \( SABC \) given that

\[
|SA|^{2} + |SB|^{2} - 5|SC|^{2} = 60
\]

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