
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 roots of the equation \( x^{4}-16 x^{3}+86 x^{2}-176 x+105=0 \) form a sequence whose eighth and thirteenth terms \([a_{8}=r\) and \(a_{13}=R]\) are equal to the radii of two spheres. If the smaller sphere is shot through the center of the larger sphere, what will be the volume of the pierced sphere and what is the surface area of the hole?

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