
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
For dessert, Melinda eats a spherical scoop of ice cream with a diameter of 2 inches. However, she prefers to eat her ice cream in cube-like shapes. She has a special machine which, given a sphere placed in space, cuts it through the planes \(x=n, y=n\), and \(z=n\) for every integer \(n\) (not necessarily positive). Melinda centers the scoop of ice cream uniformly at random inside the cube \(0 \leq x, y, z \leq 1\), and then cuts it into pieces using her machine. What is the expected number of pieces she cuts the ice cream into?

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