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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