12.3.1.8 Algebraic

Problem: Find the sum of natural values of the parameter \( a \), for each of which the system has at least one solution \[ \left\{\begin{array}{c} \sqrt{(x-a)^{2}+(y+3)^{2}}+\sqrt{(x+2)^{2}+(y-a)^{2}}= \\ =\sqrt{2 a^{2}+10 a+13} \\ y=a^{2}-4 a-6 \end{array}\right. \]