Let the domain of the function \( f(x) = \log_{3}\!\big(\log_{5}\!\big(\log_{7}(9x - x^{2} - 13)\big)\big) \) 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:
