
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
A regular triangular pyramid \( SABC \) is given, where \( S \) is its apex. The edge \( SC \) of the pyramid coincides with the lateral edge of a regular triangular prism \( A_1 B_1 C A_2 B_2 S \) (with \( A_1 A_2 \), \( B_1 B_2 \), and \( CS \) as its lateral edges, and \( A_1 B_1 C \) as one of its bases). The vertices \( A_1 \) and \( B_1 \) of the prism lie in the plane of the face \( SAB \) of the pyramid. What fraction of the volume of the entire pyramid is occupied by the part of the pyramid that lies inside the prism, given that the ratio of the length of the lateral edge of the prism to the length of the side of its base is \( \frac{2}{\sqrt{3}} \)?

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