2019 NYJC P1 Q9

The complex numbers \(z\) and \(w\) satisfy the simultaneous equations

\(\left| z \right|-{{w}^{*}}=-3-\sqrt{2}\mathbf{i}\) and \({{w}^{*}}+w+5z=1+20\mathbf{i}\),

where \({{w}^{*}}\) is the complex conjugate of \(w\). Find the value of \(z\) and the corresponding value of \(w\).[4]

It is given that \(8\mathbf{i}\) is a root of the equation \(\mathbf{i}{{z}^{3}}+\left( 8-2\mathbf{i} \right){{z}^{2}}+az+40=0\) where \(a\) is a complex number.