\[ \begin{cases} a + 3 b + 2 ab = 12.965 \\ 5 b + 2 c + 9 bc = 111.375 \\ 4 c + 4 a + 10 ac = 73.37 \end{cases} \]

The above nonlinear system of equations in \(a\), \(b\), and \(c\) has a unique solution with \( a \gt 0\), \(b \gt 0\), and \(c \gt 0 \). Find this solution numerically, and as your answer enter the value of \( 10(a + b + c) \).

**Hint**: You can use the multivariate Newton-Raphson method. It is very fast and efficient..

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