Let the domain of the function \(f(x)={\log }_{3}{\log }_{5}{\log }_{7}\left(9x-{x}^{2}-13\right)\) be the interval \((m…

Let the domain of the function \(f(x)={\log }_{3}{\log }_{5}{\log }_{7}\left(9x-{x}^{2}-13\right)\) be the interval \((m, n)\). Let the hyperbola \(\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\) have eccentricity \(\frac{n}{3}\) and the length of the latus rectum \(\frac{8m}{3}\). Then \({b}^{2}-{a}^{2}\) is equal to: