Ellipsoid with pi

The symmetric matrix $A$ of the quadratic form $q(x,y,z)=3x^2+116y^2+4xy+z^2=\frac{7}{\pi}$ is $A=\begin{bmatrix}3&2&0\\2&116&0\\0&0&1\end{bmatrix}$ with $\det A=344$ . If $\lambda_k$ , for $k=1,2,3$ , are the eigenvalues of $A$ , then the semi-axes of the ellipsoid $q(x,y,z)=\frac{7}{\pi}$ are $c_k=\sqrt{\frac{7}{\lambda_k\pi}}$ and the volume of the ellipsoid is $V=\frac{4\pi}{3}c_1c_2c_3=\frac{4\pi}{3}\sqrt{\frac{7^3}{\pi^3\det A}}=\frac{7}{3}\sqrt{\frac{14}{43\pi}}$ , so that the sum we seek is $\boxed{67}$