1. Let $\triangle A B C$ have internal angles $A, B, C$ with opposite sides $a, b, c$ respectively. Let vectors $\boldsymbol{m}=(\sin A, b+c)$ and $\boldsymbol{n}=(\sin C-\sin B, a-b)$, and there exists a real number $\lambda$ such that $\boldsymbol{m}=\lambda \boldsymbol{n}$.
(1) Find the size of angle $C$;
(2) If $a+b=k c$, find the range of the real number $k$.
(1) $\boldsymbol{m} / / \boldsymbol{n} \Rightarrow a(a-b)=(b+c)(c-b) \Rightarrow a^{2}-a b=c^{2}-b^{2} \Rightarrow c^{2}=a^{2}+b^{2}-a b$
$$
\Rightarrow \cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}=\frac{1}{2} \Rightarrow C=60^{\circ} \text {. }
$$
(2) $k=\frac{a+b}{c}=\frac{\sin A+\sin B}{\sin C}=\frac{2 \sqrt{3}}{3}\left[\sin A+\sin \left(120^{\circ}-A\right)\right]$
$$
=\frac{2 \sqrt{3}}{3}\left(\frac{3}{2} \sin A+\frac{\sqrt{3}}{2} \cos A\right)=2\left(\frac{\sqrt{3}}{2} \sin A+\frac{1}{2} \cos A\right)=2 \sin \left(A+30^{\circ}\right) \text {, }
$$

Since $30^{\circ}<A+30^{\circ}<150^{\circ}$, so $k=2 \sin \left(A+30^{\circ}\right) \in(1,2]$.