一个兔子和一个猎人在平面上玩一个游戏, 兔子的出发点 \( A_0 \) 和 猎人的出发点 \( B_0 \) 相同. \( n-1 \) 轮游戏后, 兔子位于 \( A_{n-1} \) 而猎人位于 \( B_{n-1} \) 处. 在第 \( n \) 轮, 规则如下 - 兔子可以选择移动至一个地点 \( A_n, \) 此位置满足 \( d(A_{n},A_{n-1})=1, \) 即两点局距为1. 且猎人不知道兔子的确切位置. - 猎人年久失修的仪表盘会显示兔子的位置 \( P_n, \) 但此仪表盘仅保证 \( A_n \) 于 \( P_n \) 距离不超过1. - 猎人可以选择移动至 \( B_n \) 且满足 \( d(B_n,B_{n-1})=1 .\) 兔子知道猎人的位置. 猎人是否有一个策略使得, 无论兔子如何移动, 无论仪表盘如何显示, 在 \( 10^9 \) 轮之后, 他可以保证他与兔子的距离小于100?

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A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit’s starting point, $A_0,$ and the hunter’s starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order: The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly$1.$ A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$