Let z be a complex number that satisfies the equation
\frac{z-4}{z^{2}-5 z+1}+\frac{2 z-4}{2 z^{2}-5 z+1}+\frac{z-2}{z^{2}-3 z+1}=\frac{3}{z}
Over all possible values of z, find the sum of the values of
\left|\frac{1}{z^{2}-5 z+1}+\frac{1}{2 z^{2}-5 z+1}+\frac{1}{z^{2}-3 z+1}\right|