
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 and \(\lambda\) a positive real number. Initially, there are \( n \) fleas on a horizontal line, not all at the same point. We define a move of choosing two fleas at some points \( A \) and \( B \), with \( A \) to the left of \( B \), and letting the flea from \( A \) jump over the flea from \( B \) to the point \( C \) such that \( \frac{BC}{AB} = \lambda \). Determine all values of \(\lambda\) such that for any point \( M \) on the line and for any initial position of the \( n \) fleas, there exists a sequence of moves that will take them all to the position right of \( M \).

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