# 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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