
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
 $F_n$  is the Fibonacci sequence  $F_0 = F_1 = 1$ ,  $F_{n+2} = F_{n+1} + F_n$ . Find all pairs  $m > k \geq 0$  such that the sequence  $x_0, x_1, x_2, ...$  defined by  $x_0 = \frac{F_k}{F_m}$  and  $x_{n+1} = \frac{2x_n - 1}{1 - x_n}$  for  $x_n \not = 1$ , or  $1$  if  $x_n = 1$ , contains the number  $1$ 

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