Let $$\vec{a_k} = (\tan\theta_k)\hat{i} + \hat{j}$$ and $$\vec{b_k} = \hat{i} - (\cot\theta_k)\hat{j}$$, where $$\theta_k = \frac{2^{k-1}\pi}{2^n + 1}$$, for some $$n \in \mathbb{N}$$, $$n > 5$$. Then the value of $$\frac{\sum_{k=1}^{n}|\vec{a_k}|^2}{\sum_{k=1}^{n}|\vec{b_k}|^2}$$ is _____.