
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
Consider the Fibonacci sequence defined by \( F_{0} = 0 \), \( F_{1} = 1 \), and for all \( n \geq 1 \), \( F_{n+1} = F_{n} + F_{n-1} \). Determine all pairs of integers \((k, m)\) satisfying \( m > k > 0 \) for which the sequence \( x_{n} \) defined by \( x_{0} = \frac{F_{k}}{F_{m}} \) and

\[ 
x_{n+1} = \frac{2 x_{n} - 1}{1 - x_{n}} \text{ if } x_{n} \neq 1, \quad x_{n+1} = 1 \text{ if } x_{n} = 1 
\]

contains the number 1.

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