Too many unknowns!

$\large{\begin{aligned} \color{#D61F06}{a_1x_1 + a_2x_2+a_3x_3+a_4x_4} &= \color{#D61F06}{1432} \\ \color{#20A900}{a_7x_1 + a_6x_2+a_5x_3+a_8x_4} &= \color{#20A900}{2341} \\ \color{#3D99F6}{a_{11}x_1 + a_{12}x_2+a_{13}x_3+a_{10}x_4} &= \color{#3D99F6}{3412} \\ \color{magenta}{a_{17}x_1 + a_{16}x_2+a_{15}x_3+a_{14}x_4} &= \color{magenta}{4321} \end{aligned} }$

Suppose $a_1 , a_2 , a_3, \ldots, a_{15} , a_{16} , a_{17}$ form an arithmetic progression such that $a_9=257$ . Find the value of $\color{#624F41}{\lfloor x_1 + x_2+x_3+x_4 \rfloor}$.