Limit, Continuity and Differentiability: JEE Main 2026 Mathematics question

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2-2f'(x)m+f''(x)=0$ in m, has two equal roots for every $x\in\mathbb{R}$. If $f(0)=1$, $f'(0)=2$, and $(\alpha,\beta)$ is the largest interval in which the function $f(\log_e x-x)$ is increasing, then $\alpha+\beta$ is equal to ______.