
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
Let \(C\) be a cube with side length 4 and center \(O\). Let \(S\) be the sphere centered at \(O\) with radius 2. Let \(A\) be one of the vertices of the cube. Let \(R\) be the set of points in \(C\) but not in \(S\), which are closer to \(A\) than to any other vertex of \(C\). Find the volume of \(R\).

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