Let $S={x\in [-\pi ,\pi ]:\sin x(\sin x+\cos x)=a,a\in Z}$. Then $n\left(S\right)$ is equal to: [JEE Main 2026, 2 Apr (Shift 1)]

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Let \(S={x\in [-\pi ,\pi ]:\sin x(\sin x+\cos x)=a,a\in Z}\). Then \(n\left(S\right)\) is equal to: [JEE Main 2026, 2 Ap…