Find the minimum value of $x+y+z$ for positive exponents $x,y,z$?

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$x,y,z$ are positive integers, for equation $$2^{3x}+2^{5y}=2^{7z}$$ find the minimum value for $x+y+z$.

I am not sure is it possible to find all $x,y,z$ pairs. If so, can you provide a strategy for it?