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This produces 3 equivalent Diophantine equations. How can we show the reduction steps? I need to do this for a paper.

Clear[x, y, z, n];
lexp = n >= 2 && x == (3^n - y)/(2^n) == (3^n - 1 - z)/(2^n - 1) && 
   0 < z < 2^n - 1 && 0 < y < 2^n // FullSimplify;
ord = {{x, y, z}, {y, z, x}, {z, x, y}};
Table[Reduce[lexp, ord[[m]], Reals, Backsubstitution -> True],
{m, 1, Length[ord]}];
% // Column
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You can use Trace[..., TraceInternal -> True] for this purpose. But it produces a lot of information. So you will need some post-processing. You may start with:

    Table[Trace[
      Reduce[lexp, ord[[m]], Reals, Backsubstitution -> True],
      And, TraceInternal -> True],
    {m, 1, Length[ord]}]
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