
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
A grid strip (one cell wide) is given, which is infinite in both directions. Two cells of the strip are traps, with $-N$ cells between them, and a grasshopper is sitting on one of these cells. On each turn, we name a natural number, after which the grasshopper jumps that number of cells to the left or right (of its own choice). For which values of $N$ can we name numbers in such a way that we can guarantee to drive the grasshopper into one of the traps, regardless of where it initially sits between the traps and no matter how it chooses its jump directions? (We can always see where the grasshopper is sitting.)

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