[Siren-Of-Step]

3. $x^3 -7x^2+6x-1 = 0$
$\sum_{k=0}^{3} = 3+a+b+c+a^2+b^2+c^2+a^3+b^3+c^3 , a+b+c = 7 $
$a^2+b^2+c^2 +2ab+2bc+2ca = (a+b+c)^2 = 49 = a^2+b^2+c^2 +2(6) , a^2+b^2+c^2 = 37$
$a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) , a^3+b^3+c^3 = 7(37-(6)) + 1 = 216$

$\sum_{k=0}^{3} a^k+b^k+c^k = 263$