Let $\lim_{n \rightarrow \infty} ( \frac{n}{\sqrt{n^4+1}} - \frac{2n}{(n^2+1)\sqrt{n^4+1}} + \frac{n}{\sqrt{n^4+16}} - \\ \frac{8n}{(n^2+4)\sqrt{n^4+16}} + \dots + \frac{n}{\sqrt{n^4+n^4}} - \frac{2n \cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}} ) \\ = \frac{\pi}{k},$ using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ is equal to _________.