
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
Let \( n \geq 2 \) be a positive integer. Initially, there are \( n \) fleas sitting on a horizontal line, not all at the same point.

For a positive real number \( \lambda \), define a move as follows:

Choose two fleas sitting at points \( A \) and \( B \) on the line, where \( A \) is to the left of \( B \);

Let the flea at point \( A \) jump to point \( C \) on the line, such that \( C \) is to the right of \( B \) and satisfies \( \frac{BC}{AB} = \lambda \). Determine all values of \( \lambda \) for which, no matter the choice of point \( M \) on the line, and no matter the initial positions of the \( n \) fleas, there exists a finite sequence of moves that results in all fleas being to the right of \( M \).

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