The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$  Find
\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\]

Let's think step by step and output the final answer within \boxed{}.
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