Let $y=y(x)$ satisfies the equation $\frac{d y}{d x}-|A|=0$, for all $x>0$ where $\left[\begin{array}{ccc}y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x}\end{array}\right]$. If $y(\pi)=\pi+2$, then the value of $\left(\frac{\pi}{2}\right)$ is:
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