Orthogonality & Least Squares · Superchallenge problem

Among all solutions of the system $x_1 + x_2 + x_3 + x_4 = 4$, $x_1 + 2x_2 + 3x_3 + 4x_4 = 12$, find the smallest possible value of $x_1^2 + x_2^2 + x_3^2 + x_4^2$.