
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
On a number line at point \( P \), a point-like grasshopper is sitting. Points 0 and 1 are traps. On each turn, we name any positive number, after which the grasshopper jumps left or right (at its own choice) a distance equal to that number. For which \( P \) can we name numbers in such a way as to guarantee trapping the grasshopper in one of the traps? (We can always see where the grasshopper is sitting.)

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