It is given that \(z=\frac{3+\lambda i}{\lambda+2i}\), where \(\lambda\) is a real constant.

(a) Find the value of \(\lambda\) for which \(\arg z=\frac14\pi\).

(b) When \(\lambda\) has the value found in 5(a), find the exact value of \(|z|\), making your method clear.