Newton's sums in reverse

Algebra Level 4

a+b+cNa4+b4+c4=32a5+b5+c5=186a6+b6+c6=803 \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+ca+b+c?

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