
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 (box) with edge lengths $2$, $3$, and $5$, along with its interior. For a real $r \geq 0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^3 + br^2 + cr + d$, where $a$, $b$, $c$, and $d$ are positive real numbers. What is $\frac{bc}{ad}?$
A) 19
B) 20.67
C) 24
D) 26
E) 38

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