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

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