
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 as 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$  so that  $ \frac {BC}{AB} \equal{} \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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