12.3.1.7 Algebraic

Problem: Find the sum of natural values of the parameter \( a \), for each of which the system has at least one solution. \[ \left\{\begin{array}{c} \sqrt{(x+a)^{2}+(y+3)^{2}}+\sqrt{(x-4)^{2}+(y-a)^{2}}= \\ =\sqrt{2 a^{2}+14 a+25} \\ y=a^{2}-5 a-7 \end{array}\right. \]