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