The solution (a) I gave is not all that different from the spirit of how one
solves the following mechanics problem in an inertial frame!!
You have a bunch of charged, massive partices, some of them hooked together
with springs and some with (massless, non-stretchable) ropes attached
between them etc etc. There is also a non-uniform magnetic field present,
agent unspecified. What is the motion of the system.
One solves this applying Newton's 2nd law, that is solve the equations of
motion! You never need to worry about using Newton's 3rd law to whatever it
was that produced the non-uniform magnetic field. You may find it helpful to
apply the 3rd law between some of the particles present in the system whose
motions you are trying to analyze.