
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
A sphere with a radius of $r = 10 \mathrm{dm}$ is divided into two parts such that the part towards the center is the mean proportional between the entire radius and the other part. A plane was placed perpendicularly to the radius at the dividing point, and a regular four-sided pyramid is constructed in the larger segment of the sphere such that the apex is on the surface of the sphere. Find the volume, surface area, and slant height angle of the pyramid.

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