
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
Given a triangular prism \( S-ABC \) with a base \( \triangle ABC \) that is an equilateral triangle, the projection of point \( A \) on the side \( SB \) is the orthocenter \( H \) of \( \triangle SBC \). The dihedral angle \( H-AB-C \) has a planar angle of \( 30^\circ \), and \( SA = 2\sqrt{3} \). Find the volume of the triangular prism \( S-ABC \).

(Note: This problem was from the 1999 National High School Mathematics League in China.)

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