
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
\( k \) is a positive real number. \( N \) is an integer greater than 1. \( N \) points are placed on a line, not all coincident. A move is carried out as follows. Pick any two points \( A \) and \( B \) which are not coincident. Suppose that \( A \) lies to the right of \( B \). Replace \( B \) by another point \( B' \) to the right of \( A \) such that \( AB' = k \cdot BA \). For what values of \( k \) can we move the points arbitrarily far to the right by repeated moves?

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