If the eccentricity \( e \) of the hyperbola \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), passing through \( (6, 4\sqrt{3}) \), satisfies \( 15(e^{2} + 1) = 34e \), then the length of the latus rectum of the hyperbola \( \frac{x^{2}}{b^{2}} - \frac{y^{2}}{2(a^{2} + 1)} = 1 \) is:
