$\begin{aligned} & |z|+z=3+i \\ & \Rightarrow \sqrt{x^2+y^2}+x+i y=3+i[\text { let } z=x+i y] \\ & \Rightarrow \sqrt{x^2+y^2}+x=3 \text { and } y=1 \\ & \Rightarrow \sqrt{x^2+1^2}+x=3 \\ & \Rightarrow x^2+1=(3-x)^2 \\ & \Rightarrow x=\frac{4}{3} \\ & \Rightarrow|z|=\sqrt{x^2+y^2}=\sqrt{\left(\frac{4}{3}\right)^2+1^2}=\frac{5}{3}\end{aligned}$