(1) For what value of the real number $a$ does the inequality system $\left\{\begin{array}{l}x^{2}+y^{2}-2 y \leqslant 1, \\ x+y+a=0\end{array}\right.$, have a unique solution, and find this solution.
(1) When $a=1$, there is a unique solution $\left\{\begin{array}{l}x=-1, \\ y=0 ;\end{array}\right.$ When $a=-3$, there is a unique solution $\left\{\begin{array}{l}x=1, \\ y=2\end{array}\right.$