
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
We call a set  $S$  a *good* set if  $S=\{x,2x,3x\}(x\neq 0).$  For a given integer  $n(n\geq 3),$  determine the largest possible number of the *good* subsets of a set containing  $n$  positive integers.

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