(a) Solve the equation \((1 + 2i)w + iw^* = 3 + 5i\). Give your answer in the form \(x + iy\), where \(x\) and \(y\) are real.

(b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z - 2 - 2i| \leq 1\) and \(\arg(z - 4i) \geq -\frac{1}{4}\pi\).

(ii) Find the least value of \(\text{Im } z\) for points in this region, giving your answer in an exact form.