
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 pyramid \(S-ABC\) with a base that is an equilateral triangle. The point \(A\)'s projection \(H\) on the side face \(SBC\) is the orthocenter of \(\triangle SBC\). The dihedral angle between the faces \(H-ABC\) has a plane angle of \(30^\circ\) and \(SA = 2\sqrt{3}\). What is the volume of the triangular pyramid \(S-ABC\)?

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