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 \( x^{2}-2\!\left(\ \displaystyle \sum_{a\in S} \alpha\right)^{2}x+\displaystyle \sum_{a\in S}(\alpha+1)^{2}\beta=0 \) has real roots, is _______.
