<s>[INST] What is the value of $a^4+b^4+c^4$? 
Consider $a,b,c$ such that $a+b+c =1, a^2+b^2+c^2=2$ and $a^3+b^3+c^3=3$. Find the value of $a^4+b^4+c^4$, if possible.

Trial: I observe that 
\begin{align}
a^4+b^4+c^4 &=(a^2+b^2+c^2)^2-2(a^2b^2+b^2c^2+c^2a^2)\\&=2^2-2[(ab+bc+ca)^2-2abc(a+b+c)]\\&=4-2[(ab+bc+ca)^2-2abc]
\end{align} 
Then 
\begin{align}
(a+b+c)^2 &=a^2+b^2+c^2+2(ab+bc+ca)\\\implies 1^2= 2+ 2(ab+bc+ca)\\\implies ab +bc+ca &=-\dfrac12
\end{align}
Then I am stuck. Please help.
 [/INST] Note that 
\begin{align}
a^3+b^3+c^3-3abc &=(a+b+c)(a^2+b^2+c^2-ab-ac-ca)\\\implies3-3abc&=1\cdot(2-(ab+bc+ca))\\\implies abc &= \dfrac16
\end{align}
</s>