
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 edge lengths $2$, $3$, and $5$, including its interior. For a real number $r \geq 0$, define $S(r)$ as the set of points in $3$-dimensional space that are at most a distance $r$ away from any 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 constants. Compute $\frac{bc}{ad}$.
- **A) 19**
- **B) 24**
- **C) 26**
- **D) 15.5**
- **E) 38**

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