
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's call the arrangement of $N$ grasshoppers in distinct points on a line $k$-lucky if the grasshoppers, by making the necessary number of moves according to the checkers' rules, can achieve that the sum of pairwise distances between them decreases by at least $k$ times. For which $N \geq 2$ is there an arrangement that is $k$-lucky for all natural $k$? (In the game, one of the grasshoppers jumps to a point symmetrical to it relative to another grasshopper.)

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