How to use Ampère's law in this problem?

I want to prove that the magnetic field inside a cylindrical cavity in a long, cylindrical conductor carrying uniformly distributed current i and having radius R is uniform. The radius of cavity is ‘a’ and it’s axis is parallel to the main conductor axis. However the axis of cavity is not coaxial with main conductor.

My attempt- I was able to think that the ampèrian loop should be rectangular with two long sides of length L and two short sides of length less than the radius of cavity. Now if one of the longer side is along the axis of cavity, the other longer side should be parallel to this and to ensure that the line integral along shorter side does not contribute significantly, we may take the longer side very long. After this I am having confusion in proving that the magnetic field at center of cavity is same as that at a off axis point. I am having confusion with directions and magnitudes of the field along sides of rectangle.