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Algebra Level 4

\[ \begin{cases} x_1 + 4 x_2 + 9 x_3 +16x_4 + 25x_5 + 36x_6+ 49 x_7 & = & 4 \\ 4x_1 + 9 x_2 + 16x_3 +25x_4 + 36x_5 + 49x_6 + 64 x_7 & = & 44 \\ 9 x_1 + 16 x_2 + 25 x_3 +36x_4 + 49x_5 + 64x_6 + 81 x_7 & = & 444 \\ \end{cases} \]

If \(x_1, x_2, \ldots , x_7\) satisfy the system of equations above, find the value of

\[ 16x_{ 1 }+25x_{ 2 }+36x_{ 3 }+49{ x }_{ 4 }+64x_5 + 81x_6 +{ 100x }_{ 7 } .\]

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