Let \(v\) and \(w\) be distinct, randomly chosen roots of the equation \(z^{1997}-1=0\). Let \(m/n\) be the probability that \(\sqrt{2+\sqrt{3}}\le |v+w|\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

(第十五届AIME1997 第14题)