Let the function $$f : [1, \infty) \to \mathbb{R}$$ be defined by

$$f(t) = \begin{cases} (-1)^{n+1} \cdot 2, & \text{if } t = 2n-1, \, n \in \mathbb{N}, \\ \frac{(2n+1-t)}{2} f(2n-1) + \frac{(t-(2n-1))}{2} f(2n+1), & \text{if } 2n-1 < t < 2n+1, \, n \in \mathbb{N}. \end{cases}$$

Define $$g(x) = \int_1^x f(t) \, dt$$, $$x \in (1, \infty)$$. Let $$\alpha$$ denote the number of solutions of the equation $$g(x) = 0$$ in the interval $$(1, 8]$$ and $$\beta = \lim_{x \to 1^+} \frac{g(x)}{x-1}$$. Then the value of $$\alpha + \beta$$ is equal to ______.