
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
Given a positive real number \( k \), for which real \( x_0 \) does the sequence \( x_0, x_1, x_2, \ldots \) defined by \( x_{n+1} = x_n (2 - k x_n) \) converge to \( \frac{1}{k} \)?

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