
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 right cylinder with a base radius of 4 units and a height of 10 units is inscribed in a sphere of radius 6 units. A right cone with the same base radius as the cylinder and height equal to half the cylinder's height is attached to one base of the cylinder. Calculate the total volume, in cubic units, of the space inside the sphere and outside the combined cylinder and cone. Give your answer in the form of $X\pi$, where $X$ is a common fraction.

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