Number Theory Inequality

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Given the set \( \{2, 3, 4, \ldots, n \}\) , we choose a subset \( 7 \) distinct integers. Given that, no matter what \( n \) is, we can find two integers \( X, Y \) in our subset such that \( 1< \frac{Y}{X} <2 \), what is the greatest possible value of \( n \)?

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