
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 base of the pyramid \( SABC \) lies an equilateral triangle \( ABC \) with side length \( 2\sqrt{3} \), and \( SA = SB = SC = \sqrt{7} \). A sphere \( S_1 \) is inscribed in the trihedral angle at vertex \( C \). A sphere \( S_2 \), whose radius is three times that of sphere \( S_1 \), touches sphere \( S_1 \), and the planes \( SAC \) and \( ABC \). Additionally, the segment of the line \( SB \) that lies inside sphere \( S_2 \) is equal to \( \frac{6}{\sqrt{7}} \). Find the radius of sphere \( S_2 \).

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