Find minimum value of \(n \ge 3\) such that among any \(n\) positive real numbers \(a_1, a_2, \ldots, a_n\) with

\[\large\ \max(a_1, a_2, \ldots, a_n) \le n \cdot \min(a_1, a_2, \ldots, a_n)\]

there exist three that are the side lengths of an acute triangle.

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