# Not your average complex problem

Algebra Level 5

It is given that for some complex numbers $$z$$, $$\left|z-5 \sqrt{3} -5i \right| = 5$$, which is equivalent to $$\left|\dfrac{1}{z}-\dfrac{1}{a\sqrt{b}}+\dfrac{i}{c} \right|=\dfrac{1}{d}$$, where $$a$$, $$b$$, $$c$$ and $$d$$ are positive integers and $$b$$ is square-free.

Find $$a+b+c+d$$.

Clarification: $$i=\sqrt{-1}$$.

Hint: Consider what happens to $$z$$ in the complex plane when its reciprocal is taken.

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