How to make NDSolve to use NSolve solution?

Questions
1

Hi,
I have an NDSolve that requires an NSolve solution at each time step:

a[t_] := A /. NSolve[x[t]*A == 1 - x[t]^2];
NDSolve[{x'[t] == 2*a[t]*x[t], x[0] == 2}, {x[t]}, {t, 0, 1}];
Plot[x[t]/.%, {t, 0, 1}]

Which should result similar to this:

NDSolve[{x'[t] == 2*(1-x[t]^2), x[0] == 2}, {x[t]}, {t, 0, 1}];
2

Try it with

a[t_] := First[A /. Solve[x[t]*A == 1 - x[t]^2]];

This gets you the actual solution instead of a solution in a list.