Let the lines $y+2 x=\sqrt{11}+7 \sqrt{7}$ and $2 y+x=2 \sqrt{11}+6 \sqrt{7}$ be normal to a circle $C:(x-h)^2+(y-k)^2=r^2$. If the line $\sqrt{11} y-3 x=\frac{5 \sqrt{77}}{3}+11$ is tangent to the circle $C$, then the value of $(5 h-8 k)^2+5 r^2$ is equal to $\_\_\_\_$ .
