
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 \( n \) be a given positive integer. \( D_n \) is the set of all distinct numbers of the form \( 2^{a} 3^{b} 5^{c} \) with \( 0 \leq a, b, c \leq n \). \( S \subseteq D_n \). If no number in \( S \) can divide another number in \( S \), find the maximum value of \( |S| \).

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