
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 positive integer  $k\geqslant 3$  is called*fibby* if there exists a positive integer  $n$  and positive integers  $d_1 < d_2  < \ldots < d_k$  with the following properties:     $\bullet$   $d_{j+2}=d_{j+1}+d_j$  for every  $j$  satisfying  $1\leqslant j \leqslant k-2$ ,   $\bullet$   $d_1, d_2, \ldots, d_k$  are divisors of  $n$ ,   $\bullet$  any other divisor of  $n$  is either less than  $d_1$  or greater than  $d_k$ .

 Find all fibby numbers.   *Proposed by Ivan Novak.*

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