Too many unknowns!

Algebra Level 5

\[\large{\begin{align} \color{red}{a_1x_1 + a_2x_2+a_3x_3+a_4x_4} &= \color{red}{1432} \\ \color{green}{a_7x_1 + a_6x_2+a_5x_3+a_8x_4} &= \color{green}{2341} \\ \color{blue}{a_{11}x_1 + a_{12}x_2+a_{13}x_3+a_{10}x_4} &= \color{blue}{3412} \\ \color{magenta}{a_{17}x_1 + a_{16}x_2+a_{15}x_3+a_{14}x_4} &= \color{magenta}{4321} \end{align} }\]

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 \(\color{brown}{\lfloor x_1 + x_2+x_3+x_4 \rfloor}\).

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