
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 sequence of integers a *Fibonacci-type sequence* if it is infinite in both ways and  $a_{n}=a_{n-1}+a_{n-2}$  for any  $n\in\mathbb{Z}$ . How many *Fibonacci-type sequences* can we find, with the property that in these sequences there are two consecutive terms, strictly positive, and less or equal than  $N$  ? (two sequences are considered to be the same if they differ only by shifting of indices)

*Proposed by I. Pevzner*

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