Let $a_1=1,\,a_2,\,a_3,\,a_4,\ldots$ be consecutive natural numbers. Then $\tan^{-1}\!\left(\dfrac{1}{1+a_1a_2}\right)+\tan^{-1}\!\left(\dfrac{1}{1+a_2a_3}\right)+\cdots+\tan^{-1}\!\left(\dfrac{1}{1+a_{2021}a_{2022}}\right)$ is equal to: