limit with double summation

Finding value of

$$\lim_{n\rightarrow \infty}\lim_{m\rightarrow \infty}\sum^{n}_{r=1}\sum^{mr}_{k=1}\frac{m^2n^2}{(m^2n^2+k^2)(n^2+r^2)}$$

what i try

$$\lim_{m\rightarrow \infty}\lim_{n\rightarrow \infty}\sum^{mr}_{k=1}\frac{m^2n^2}{m^2n^2+k^2}\cdot \frac{1}{n}\sum^{n}_{r=1}\frac{n}{n^2+r^2}$$

$$\lim_{n\rightarrow \infty}\sum^{n}_{r=1}\frac{n}{n^2+r^2}=\int^{1}_{0}\frac{1}{1+x^2}dx = \frac{\pi}{4}$$ How do i solve first summation

help me please