Real numbers \(a\) and \(b\) are chosen 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 \(1\). The smallest possible value of \(b\) is given in the form \(\frac{k+\sqrt{m}}{n}\). Please find the value of \(k+m+n\).

Let's think step by step and output the final answer within \boxed{}.
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