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Let $\mathscr{P}=\left\{P_1, P_2, \ldots, P_{1997}\right\}$ be a set of 1997 points inside the unit circle with center at $P_1$. For each $k=1,2, \ldots, 1997$, let $x_k$ be the distance from $P_k$ to the closest point in $\mathscr{P}$ different from $P_k$. Prove that
$x_1^2+x_2^2+\cdots+x_{1997}^2 \leq 9$
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