Limit, Continuity and Differentiability: JEE Main 2025 Mathematics question

Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2 \log_e (x - 2) - x^2 + a x + 1$ is strictly increasing and $(b, c)$ be the largest open interval in which the function $g(x) = (x - 1)^3 (x + 2 - a)^2$ is strictly decreasing. Then $100(a + b - c)$ is equal to: