Newton's sums in reverse

Algebra Level 4

\[ \begin{array} { l l} a + b + c & \in \mathbb{N} \\ a^4+b^4+c^4 & =32\\ a^5+b^5+c^5& =186\\ a^6+b^6+c^6& =803\\ \end{array} \]

What is the value of \(a+b+c\)?

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