Let y=y(x) be the solution of the differential equation ((x+2) e^((y+1)/(x+2))+(y+1)) d x=(x+2) d y, y(1)=1. If the domain of y=y(x) is an open…

Let $\displaystyle y=y(x)$ be the solution of the differential equation $$\left((x+2) \mathrm{e}^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right) \mathrm{d} x=(x+2) \mathrm{d} y, y(1)=1 \text {. If the domain of } y=y(x) \text { is an open } $$ interval $\displaystyle (\alpha, \beta)$, then $\displaystyle |\alpha+\beta|$ is equal to \_\_\_\_.