Exponents are not so straightforward!

Algebra Level 5

Let \(x_0\) be the number of positive real solutions to \[\large x^{1.2^{1.2}} = 1.2^x\] And \(y_0\) be the number of positive real solutions to \[\large y^{x_0^{x_0}} =x_0^y\] Find the sum of all real positive solutions \(z\) to the equation \[\large z^{y_0}=x_0^z\]


Inspiration.

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