
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 base of the triangular pyramid \( SABC \) is an equilateral triangle \( ABC \) with a side length of 4. It is known that for any point \( M \) on the extension of the height of the pyramid \( SH \) (point \( S \) is between points \( M \) and \( H \)), the angles \( MSA, MSB, MSC, ASB, ASC \), and \( BSC \) are equal to each other. A sphere with radius 1 and centered at point \( S \) is constructed. Find the volume of the common part of the pyramid \( SABC \) and the sphere (the volume of a sphere with radius \( R \) is calculated using the formula \( V = \frac{4}{3} \pi R^3 \)).

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