1.
Let S={a : log⁡2(92a−4+1)−log⁡2(32a−4+1)=2}S = \{a \, : \, \log_2(9^{2a-4}+1) - \log_2(3^{2a-4}+1) = 2\}. Then the maximum value of β\beta for which the equation
x2−2(∑a∈Sa)x+∑a∈S(a+1)2 2β=0x^2 - 2\left(\sum_{a \in S} a\right) x + \sum_{a \in S} (a+1)^2 \, 2^{\beta} = 0

has real roots, is