
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
Let  $f_0, f_1,...$  be the Fibonacci sequence:  $f_0 = f_1 = 1, f_n = f_{n-1} + f_{n-2}$  if  $n \geq 2$ .
Determine all possible positive integers  $n$  so that there is a positive integer  $a$  such that $f_n \leq a \leq f_{n+1}$  and that  $a(  \frac{1}{f_1}+\frac{1}{f_1f_2}+\cdots+\frac{1}{f_1f_2...f_n}  )$ is an integer.

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