Mathematics Question from JEE Main 2023 · 25 January, Shift 1

Let $S = \left\{\alpha : \log_2 \left(9^{2\alpha-4} + 13\right) - \log_2 \left(\frac{5}{2} \cdot 3^{2\alpha-4} + 1\right) = 2 \right\}.$ Then the maximum value of $\beta$ for which the equation

$$x^2 - 2\left(\sum_{\alpha \in S} \alpha\right)^2 x + \sum_{\alpha \in S} (\alpha + 1)^2 \beta = 0$$

has real roots, is _____.