Differential Equations: JEE Main 2024 Mathematics question

If the solution curve $y=y(x)$ of the differential equation $\left(1+y^2\right)\left(1+\log _{\mathrm{e}} x\right) d x+x d y=0, x > 0$ passes through the point $(1,1)$ and $y(e)=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan \left(\frac{3}{2}\right)}$, then $\alpha+2 \beta$ is _________.