Complex Numbers

Let $\mathrm{S}=\{z: 3 \leqslant|2 z-3(1+\mathrm{i})| \leqslant 7\}$ be a set of complex numbers.

Then $\operatorname{Min}_{z \in S}\left|\left(z+\frac{1}{2}(5+3 i)\right)\right|$ is equal to :