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