
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 \( T \) be the set of numbers of the form \( 2^{a} 3^{b} \) where \( a \) and \( b \) are integers satisfying \( 0 \leq a, b \leq 5 \). How many subsets \( S \) of \( T \) have the property that if \( n \) is in \( S \) then all positive integer divisors of \( n \) are in \( S \)?

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