
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  $f_n=\left(1+\frac{1}{n}\right)^n\left((2n-1)!F_n\right)^{\frac{1}{n}}$ . Find  $\lim \limits_{n \to \infty}(f_{n+1} - f_n)$  where  $F_n$  denotes the  $n$ th Fibonacci number (given by  $F_0 = 0$ ,  $F_1 = 1$ , and by  $F_{n+1} = F_n + F_{n-1}$  for all  $n \geq 1$ 

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