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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