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 $\mathrm { x } ^ { 2 } - 2 \left( \sum _ { \alpha \in s } \alpha \right) ^ { 2 } \mathrm { x } + \sum _ { a \in s } ( \alpha + 1 ) ^ { 2 } \beta = 0$ has real roots, is $\_\_\_\_$ .