Let \( I(x) = \int \frac{6}{\sin^2 x \, (1-\cot x)^2} \, dx \). If \( I(0) = 3 \), then \( I\!\left(\frac{\pi}{12}\right) \) is equal to:
\( 2\sqrt{3} \)
\( 3\sqrt{3} \)
\( \sqrt{3} \)
\( 6\sqrt{3} \)
