Can I solve this other than using Newton's sums or Newton's identities? 
$\begin{cases}
 &x + y + z = 12\\& x^2 + y^2 + z^2 =  12
\\ & x^3 +y^3 + z^3 = 12
\end{cases}$
If $x,$ $y,$ and $z$ satisfy the system of equations above, what is the value of $x^4+y^4+z^4?$ 

People told me that this can be solved by using Newton's sums or Newton's identities which I don't know how. Does anybody know how to do that or any other method to solve this? 

Let $x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.
Thus, $$u=4,$$
$$3v^2=\frac{12^2-12}{2}=66,$$ which gives $$v^2=22.$$
Also, $$12=x^3+y^3+z^3=27u^3-27uv^2+3w^3,$$ which gives $$w^3=220$$ and use
$$x^4+y^4+z^4=81u^4-108u^2v^2+18v^4+12uw^3.$$
I got $1992.$
I used the known $uvw$'s substitutions: https://artofproblemsolving.com/community/c6h278791
