
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
The chords \(A A'\), \(B B'\), and \(C C'\) of a sphere intersect at a common point \(S\). Find the sum \(S A' + S B' + S C'\), given \(A S = 6\), \(B S = 3\), \(C S = 2\), and the volumes of the pyramids \(SA B C\) and \(S A' B' C'\) are in the ratio \(2: 9\). If the answer is not an integer, round it to the nearest hundredth.

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