
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 right circular cylinder with height $4$ and radius $2$, along with its interior. For a real $r \geq 0$, let $T(r)$ denote the set of points in 3-dimensional space that are within a distance $r$ of some point in $C$. Additionally, at each circular end of the cylinder, a square pyramid with a height equal to $r$ and base side equal to $4r$ adds to the set $T(r)$. The volume of $T(r)$ can be expressed as $ar^{3} + br^{2} + cr + d$, where $a, b, c, and d$ are constants. Determine $\frac{bc}{ad}$.
A) $\frac{3}{8}$
B) $\frac{5}{2}$
C) $\frac{9}{4}$
D) $\frac{11}{4}$
E) $6$

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