
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 integer \( n \geq 2 \), initially there are \( n \) fleas located on a line such that they are not all at the same point. For any given positive real number \( \lambda \), a "move" can be defined as follows:
1. Select any two fleas located at points \( A \) and \( B \), with \( A \) to the left of \( B \);
2. Let the flea at point \( A \) jump to a point \( C \) on the line to the right of \( B \), such that \( \frac{BC}{AB} = \lambda \).

Determine all possible positive real numbers \( \lambda \) such that for any given point \( M \) on the line and any initial positions of the \( n \) fleas, it is always possible to move all fleas to the right of point \( M \) after a finite number of moves.

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