
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
Consider a right rectangular prism \(B\) with edge lengths \(2,\ 5,\) and \(6\), including its interior. For any real \(r \geq 0\), let \(T(r)\) be the set of points in 3D space within a distance \(r\) from some point in \(B\). The volume of \(T(r)\) is expressed as \(ar^{3} + br^{2} + cr + d\), where \(a,\) \(b,\) \(c,\) and \(d\) are positive real numbers. Calculate \(\frac{bc}{ad}\).
A) $\frac{1560}{60}$
B) $\frac{8112}{240}$
C) $\frac{3792}{120}$
D) $\frac{6084}{180}$

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