If $y=y(x)$ is the solution of the differential equation $\left(1+e^{2 x}\right) \frac{d y}{d x}+2\left(1+y^2\right) e^x=0$ and $y(0)=0$, then $6\left(y^{\prime}(0)+\left(y\left(\log _e \sqrt{3}\right)\right)^2\right)$ is equal to:
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