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 \( \dfrac{\sum_{k=1}^{n}|\vec{a}_{k}|^{2}}{\sum_{k=1}^{n}|\vec{b}_{k}|^{2}} \) is ________.
