2023 SSMO Accuracy Round Problems/Problem 3

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Problem

Suppose that $a, b, c$ are real numbers such \begin{align*}     a + b - c &= 4 \\     a^2 + b^2 + c^2 &= 14 \\     a^3 + b^3 - c^3 &= 34 \\ \end{align*} Find the sum of all possible values of $a+b+c$.

Solution