
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 set of positive integers is said to be *pilak* if it can be partitioned into 2 disjoint subsets  $F$  and  $T$ , each with at least  $2$  elements, such that the elements of  $F$  are consecutive Fibonacci numbers, and the elements of  $T$  are consecutive triangular numbers. Find all positive integers  $n$  such that the set containing all the positive divisors of  $n$  except  $n$  itself is pilak.

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