Let $(x_1,y_1),$ $(x_2,y_2),$ $\dots,$ $(x_n,y_n)$ be the real solutions to \[x^2 + y^2 = 6y - 4x + 20 \quad \text{and} \quad 4y = x^2 + 4x + 2\] Find $x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.$
+1364 
