You really only have one function.
is the same as
and
is the same as
The first of those is 0 < x < 12.5, and the second is 0 < x < 1600000, so the first one of those dominates.
You have
which you can now rewrite as
0 < 176^a-0.0003^(1-a) < 12.5
The left boundary is reached at a = log(3/10000)/log(33/625) which is about 2.75792414361621 and the right boundary is reached at a approximately 0.4887342220443237643
But 0.488 etc is less than the lower bound of the constraint 0.5 < a < 0.75 so there is no solution to the equation.
You can plot (176^a-0.0003^(1-a))*0.016 over 0.5 to 0.75 and you will see that the value is everywhere greater than 0.2 which is the upper bound for z.
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