If the solution curve \( y=y(x) \) of the differential equation \( (1+y^{2})(1+\log_{e} x)\,dx + x\,dy = 0,\ x>0 \) 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 ____________.
