An unusual limit of lengths of intervals

Calculus Level 5

For a positive integer \(n\), let \(S_n\) be the total sum of the intervals of \(x\) such that \(\sin 4n x\geq \sin x\), and \(0\leq x\leq \frac{\pi}{2}.\)

If \(\displaystyle \lim_{n\to\infty} S_n= \frac{a \pi }{b},\) where \(a,b\) are coprime integers. Find \(a+b\).

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