
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 \( B \) be a right rectangular prism with edges lengths \( 2, 4, \) and \( 5 \), together with its interior. For real \( r \geq 0 \), let \( S(r) \) be the set of points in 3-dimensional space within distance \( r \) of some point in \( B \). The volume of \( S(r) \) is expressed as \( ar^3 + br^2 + cr + d \). Find \( \frac{bc}{ad} \).

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