
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
In a regular triangular pyramid (tetrahedron), there is a sphere with a radius of 1. At the midpoint of the height of the pyramid, the sphere externally touches a hemisphere. The hemisphere is supported by a circle inscribed in the base of the pyramid, and the sphere touches the lateral faces of the pyramid. Find the lateral surface area of the pyramid and the angle between the lateral faces of the pyramid.

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