Scipy's parameterization of Generalized Gamma vs wikipedia's

Question

Wikipedia's parameterization of the Generalized Gamma distributions pdf is relatively straightforward

$$ f(t; a,d,p) = \frac{p/a^d}{\Gamma(d/p)} t^{d-1} e^{-(t/a)^p} $$

Scipy's parameterization of the pdf is $$ f(t; a,c) = \frac{|c| t^{ca - 1}}{\Gamma(a)} e^{-t^c} $$

Im having trouble understanding where Scipy's parameterization came from. I attempted to prove the two were equivalent with algebra but I ended up with some leftover terms. How does the math work out and how do the parameters of each relate to eachother between the 2 parameterizations?

Attempt below

$$ f(t; a,c) = f(t; z,c) = \frac{|c| t^{cz - 1}}{\Gamma(z)} e^{-t^c}\\ \frac{|c| t^{cz - 1}}{\Gamma(z)} e^{-t^c} = \frac{p/a^d}{\Gamma(d/p)} t^{d-1} e^{-(t/a)^p}\\ \text{guess substitution:}\:\: z = d/p\\ \frac{|c| t^{c(d/p) - 1}}{\Gamma(d/p)} e^{-t^c} = \frac{p/a^d}{\Gamma(d/p)} t^{d-1} e^{-(t/a)^p}\\ = |c| t^{c(d/p) - 1} e^{-t^c} = p/a^d t^{d-1} e^{-(t/a)^p}\\ \text{guess substitution:}\:\: c = p\\ |(p)| t^{(p)(d/p) - 1} e^{-t^p} = p/a^d t^{d-1} e^{-(t/a)^p}\\ = t^{d - 1} e^{-t^p} = p/a^d t^{d-1} e^{-(t/a)^p}\\ = e^{-t^p} = p/a^d e^{-(t/a)^p}\\ $$

and this last line is where I've gotten stuck.