
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
A four-element set  $\{a, b, c, d\}$  of positive integers is called *good* if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set  $\{2, 4, 6, 8\}$  is good since the greatest common divisor of  $2$  and  $6$  is  $2$ , and it divides  $4\times 8=32$ .

Find the greatest possible value of  $n$ , such that any four-element set with elements less than or equal to  $n$  is good.

*Proposed by Victor and Isaías de la Fuente*

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