
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 hunter and an invisible rabbit are playing a game on the Euclidean plane. The initial position of the rabbit $A_0$ and the initial position of the hunter $B_0$ coincide. After $n-1$ rounds of the game, the rabbit is located at point $A_{n-1}$, and the hunter is located at point $B_{n-1}$. In the $n$-th round, the following events occur in sequence:
1. The rabbit moves invisibly to point $A_n$, such that the distance between points $A_{n-1}$ and $A_n$ is exactly 1;
2. A locating device feeds back a point $P_n$ to the hunter, and the device can only guarantee that the distance between point $P_n$ and $A_n$ is at most 1;
3. The hunter moves visibly to point $B_n$, such that the distance between points $B_{n-1}$ and $B_n$ is exactly 1.

The question is: Regardless of how the rabbit moves and regardless of the points fed back by the locating device, can the hunter always choose his movement in such a way that after $10^9$ rounds, he can ensure that the distance between himself and the rabbit is at most 100?

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