Let \(D\) be the domain of the function \(f(x)=\sin ^{-1}\) \(\left(\log _{3 x}\left(\frac{6+2 \log _3 x}{-5 x}\right)\r…

Let \(D\) be the domain of the function \(f(x)=\sin ^{-1}\) \(\left(\log _{3 x}\left(\frac{6+2 \log _3 x}{-5 x}\right)\right)\). If the range of the function \(g: D \rightarrow R\) defined by \(g(x)=x-[x]\), (\([x]\) is the greatest integer function) is \((\alpha, \beta)\), then \(\alpha^2+\frac{5}{\beta}\) is equal to

[JEE Main 2023, 12 Apr (Shift 1)]