For \( I(x)=\displaystyle \int \frac{\sec^2 x-2022}{\sin^{2022}x}\,dx \), if \( I\!\left(\dfrac{\pi}{4}\right)=2^{1011} \), then
\( 3^{1010}I\!\left(\dfrac{\pi}{3}\right)-I\!\left(\dfrac{\pi}{6}\right)=0 \)
\( 3^{1010}I\!\left(\dfrac{\pi}{6}\right)-I\!\left(\dfrac{\pi}{3}\right)=0 \)
\( 3^{1011}I\!\left(\dfrac{\pi}{3}\right)-I\!\left(\dfrac{\pi}{6}\right)=0 \)
\( 3^{1011}I\!\left(\dfrac{\pi}{6}\right)-I\!\left(\dfrac{\pi}{3}\right)=0 \)
