\[ \large \begin{cases} {a+b+c=7} \\ {a^2+b^2+c^2= 23} \\ {\frac1{a+b} +\frac1{b+c} + \frac1{a+c} = 31 } \end{cases} \]

If \(a,b\) and \(c\) satisfy the system of equations above, find the value of \( a^3 + b^3 + c^3 \).

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