
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 $S-ABC$ whose base is an equilateral triangle, the projection of point $A$ on the face $SBC$ is the orthocenter $H$ of triangle $\triangle SBC$. The dihedral angle between the planes $H-AB-C$ is $30^{\circ}$, and $SA = 2\sqrt{3}$. Find the volume of the triangular pyramid $S-ABC$.

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