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Find all points at which $f(z)=z^5(4\overline{z}+i|z|^2-(\text{Im }z)(\text{Re }z)^2)$ is differentiable.

Of course, we can write $z=x+iy$ and check the Cauchy-Riemann equations. But this is just too tedious. I wonder if there is a more clever way to do this.

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