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Algebra Level 5

There is no solution to the equations

\(57x+31y+19z=320\\ 48x+25y+15z=107\\ ax+by+cz=abc\)

for \(x,y\) and \(z\), where \(x, y\) and \(z\) are real numbers and \(a, b\) and \(c\) are positive integers less than 10.

Find the value of \(a+b+c\).

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