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(A) Given the system of equations: $1) \sin \left(\frac{x+y}{2}\right) = 0$ $\Rightarrow \frac{x+y}{2} = n\pi$ $\Rightarrow x+y = 2n\pi$,where $n \in

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$2$

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(A) Given the system of equations: $1) \sin \left(\frac{x+y}{2}ight) = 0$ $\Rightarrow \frac{x+y}{2} = n\pi$ $\Rightarrow x+y = 2n\pi$,where $n \in \mathbb{Z}$. $2) |x| + |y| = 1$. For $n=0$,we have $x+y=0$,which is a line passing through the origin. The graph of $|x| + |y| = 1$ is a square with vertices at $(1,0), (0,1), (-1,0), (0,-1)$. The line $x+y=0$ intersects the sides of this square at two points: $(-0.5, 0.5)$ and $(0.5, -0.5)$. For any other integer $n 
eq 0$,$x+y = 2n\pi$ implies $|x+y| = |2n\pi| \geq 2\pi \approx 6.28$. However,for the square $|x| + |y| = 1$,the maximum value of $|x+y|$ is $|x| + |y| = 1$. Since $1 Thus,there are exactly $2$ solutions.

Find the number of solutions to the system of equations $\sin \left(\frac{x+y}{2}\right)=0$ and $|x| + |y| = 1$.

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