Determine the values of \(k\) so that the inequality \(\displaystyle{\sin x \cos x\geq k}\) is satisfied \(\forall x \in \mathbb{R}\).

If \(k^2\) can be written as \(\displaystyle \frac{a}{b}\), where \(a\) and \(b\) are relatively prime integers, find \(a+b\).

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