Let z=a+b i be the complex number with |z|=5 and b>0 such that the distance between (1+2 i) z^{3} and z^{5} is maximized, and let z^{4}=c+d i. Find c+d.
Let z=a+b i be the complex number with |z|=5 and b>0 such that the distance between (1+2 i) z^{3} and z^{5} is maximized, and let z^{4}=c+d i. Find c+d.