Consider real numbers \(a\) and \(b\) with \(1 < a < b\) such that no triangle with positive area has side lengths \(1, a,\) and \(b\) or \(\frac{1}{b}, \frac{1}{a},\) and \(1\). What is the smallest possible value of \(b\)? Let's think step by step and output the final answer within \boxed{}.
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