
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 \( d(n) \) denote the number of positive integers that divide \( n \), including 1 and \( n \). Let
\[
f(n) = \frac{d(n)}{\sqrt[3]{n}}
\]
where \( n \) is of the form \( 2^a \cdot 3^b \cdot 5^c \). Find the values of \( a, b, c \) such that \( f(n) \) is maximized for this form of \( n \). Then, provide the sum of the digits of this optimal \( n \).
A) 16
B) 17
C) 18
D) 19
E) 20

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