Trigonometric Equations

If \(n\) is the number of solutions of the equation \(2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]\) and \(S\) is the sum of all these solutions, then the ordered pair \((n, S)\) is