Let \( a_{1}=1,\, a_{2}, a_{3}, a_{4}, \ldots \) be consecutive natural numbers.
Then \( \tan^{-1}\!\left(\dfrac{1}{1+a_{1}a_{2}}\right)+\tan^{-1}\!\left(\dfrac{1}{1+a_{2}a_{3}}\right)+\ldots+\tan^{-1}\!\left(\dfrac{1}{1+a_{2021}a_{2022}}\right) \) is equal to:
