An ellipse E: x^2/a^2+y^2/b^2=1 passes through the vertices of the hyperbola H: x^2/49-y^2/64=-1. Let the major and minor axes of the ellipse E…

An ellipse $\displaystyle E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $\displaystyle H: \frac{x^2}{49}-\frac{y^2}{64}=-1$. Let the major and minor axes of the ellipse $\displaystyle E$ coincide with the transverse and conjugate axes of the hyperbola $\displaystyle H$, respectively. Let the product of the eccentricities of $\displaystyle E$ and $\displaystyle H$ be $\displaystyle \frac{1}{2}$. If $\displaystyle l$ is the length of the latus rectum of the ellipse $\displaystyle E$, then the value of $\displaystyle 113 l$ is equal to \_\_\_\_.