For given \(k\in \mathbb{N}\), determine the minimum natural number \(n\) satisfying the following: no matter how one colors each number in \(\{1,2,\dots, n\}\) red or blue, there always exists (not necessarily distinct) numbers \(x_0, x_1,\dots, x_k \in [n]\) with the same color satisfying \(x_1+\dots + x_k = x_0\).

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Other solutions were submitted by 고성훈 (수리과학과 2018학번, +3), 김기수 (수리과학과 2018학번, +3).